Modulated guided surface waves

ABSTRACT

Disclosed are various systems and methods directed to the launching of a Zenneck surface wave embodying a modulated signal using a guided surface waveguide probe. A modulated signal is generated and coupled to a guided surface waveguide probe. A resulting Zenneck surface wave is launched that decays exponentially as a function of distance.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation application of U.S. application Ser.No. 14/838,852, filed on Aug. 28, 2015 entitled “MODULATED GUIDEDSURFACE WAVES”, which application claims the benefit of, and priorityto, co-pending U.S. Provisional Patent Application No. 62/049,302entitled “AMPLITUDE MODULATED GUIDED SURFACE WAVES” filed on Sep. 11,2014, both of which are incorporated herein by reference in theirentirety.

BACKGROUND

For over a century, signals transmitted by radio waves involvedradiation fields launched using conventional antenna structures. Incontrast to radio science, electrical power distribution systems in thelast century involved the transmission of energy guided along electricalconductors. This understanding of the distinction between radiofrequency (RF) and power transmission has existed since the early1900's.

BRIEF DESCRIPTION OF THE DRAWINGS

Many aspects of the present disclosure can be better understood withreference to the following drawings. The components in the drawings arenot necessarily to scale, emphasis instead being placed upon clearlyillustrating the principles of the disclosure. Moreover, in thedrawings, like reference numerals designate corresponding partsthroughout the several views.

FIG. 1 is a chart that depicts field strength as a function of distancefor a guided electromagnetic field and a radiated electromagnetic field.

FIG. 2 is a drawing that illustrates a propagation interface with tworegions employed for transmission of a guided surface wave according tovarious embodiments of the present disclosure.

FIG. 3 is a drawing that illustrates a guided surface waveguide probedisposed with respect to a propagation interface of FIG. 2 according tovarious embodiments of the present disclosure.

FIG. 4 is a plot of an example of the magnitudes of close-in and far-outasymptotes of first order Hankel functions according to variousembodiments of the present disclosure.

FIGS. 5A and 5B are drawings that illustrate a complex angle ofincidence of an electric field synthesized by a guided surface waveguideprobe according to various embodiments of the present disclosure.

FIG. 6 is a graphical representation illustrating the effect ofelevation of a charge terminal on the location where the electric fieldof FIG. 5A intersects with the lossy conducting medium at a Brewsterangle according to various embodiments of the present disclosure.

FIG. 7 is a graphical representation of an example of a guided surfacewaveguide probe according to various embodiments of the presentdisclosure.

FIGS. 8A through 8C are graphical representations illustrating examplesof equivalent image plane models of the guided surface waveguide probeof FIGS. 3 and 7 according to various embodiments of the presentdisclosure.

FIGS. 9A and 9B are graphical representations illustrating examples ofsingle-wire transmission line and classic transmission line models ofthe equivalent image plane models of FIGS. 8B and 8C according tovarious embodiments of the present disclosure.

FIG. 10 is a flow chart illustrating an example of adjusting a guidedsurface waveguide probe of FIGS. 3 and 7 to launch a guided surface wavealong the surface of a lossy conducting medium according to variousembodiments of the present disclosure.

FIG. 11 is a plot illustrating an example of the relationship between awave tilt angle and the phase delay of a guided surface waveguide probeof FIGS. 3 and 7 according to various embodiments of the presentdisclosure.

FIG. 12 is a drawing that illustrates an example of a guided surfacewaveguide probe according to various embodiments of the presentdisclosure.

FIG. 13 is a graphical representation illustrating the incidence of asynthesized electric field at a complex Brewster angle to match theguided surface waveguide mode at the Hankel crossover distance accordingto various embodiments of the present disclosure.

FIG. 14 is a graphical representation of an example of a guided surfacewaveguide probe of FIG. 12 according to various embodiments of thepresent disclosure.

FIG. 15A includes plots of an example of the imaginary and real parts ofa phase delay (Φ_(U)) of a charge terminal T₁ of a guided surfacewaveguide probe according to various embodiments of the presentdisclosure.

FIG. 15B is a schematic diagram of the guided surface waveguide probe ofFIG. 14 according to various embodiments of the present disclosure.

FIG. 16 is a drawing that illustrates an example of a guided surfacewaveguide probe according to various embodiments of the presentdisclosure.

FIG. 17 is a graphical representation of an example of a guided surfacewaveguide probe of FIG. 16 according to various embodiments of thepresent disclosure.

FIGS. 18A through 18C depict examples of receiving structures that canbe employed to receive energy transmitted in the form of a guidedsurface wave launched by a guided surface waveguide probe according tothe various embodiments of the present disclosure.

FIG. 18D is a flow chart illustrating an example of adjusting areceiving structure according to various embodiments of the presentdisclosure.

FIG. 19 depicts an example of an additional receiving structure that canbe employed to receive energy transmitted in the form of a guidedsurface wave launched by a guided surface waveguide probe according tothe various embodiments of the present disclosure.

FIGS. 20A-20E depict schematic symbols of guided surface waveguideprobes and guided surface wave receiver structures according to variousembodiments of the present disclosure.

FIGS. 21-25 depict schematic diagrams of examples of AmplitudeModulation (AM) transmission systems according to various embodiments ofthe present disclosure.

FIG. 26 is a drawing that depicts skywave and ground wave propagationthat occurs in conventional radio transmission.

FIG. 27 is a drawing of adjacent Amplitude Modulation transmissionsystems according to various embodiments of the present disclosure.

FIG. 28 is a drawing of adjacent areas served by respective AmplitudeModulation transmission systems according to various embodiments of thepresent disclosure.

DETAILED DESCRIPTION

To begin, some terminology shall be established to provide clarity inthe discussion of concepts to follow. First, as contemplated herein, aformal distinction is drawn between radiated electromagnetic fields andguided electromagnetic fields.

As contemplated herein, a radiated electromagnetic field compriseselectromagnetic energy that is emitted from a source structure in theform of waves that are not bound to a waveguide. For example, a radiatedelectromagnetic field is generally a field that leaves an electricstructure such as an antenna and propagates through the atmosphere orother medium and is not bound to any waveguide structure. Once radiatedelectromagnetic waves leave an electric structure such as an antenna,they continue to propagate in the medium of propagation (such as air)independent of their source until they dissipate regardless of whetherthe source continues to operate. Once electromagnetic waves areradiated, they are not recoverable unless intercepted, and, if notintercepted, the energy inherent in the radiated electromagnetic wavesis lost forever. Electrical structures such as antennas are designed toradiate electromagnetic fields by maximizing the ratio of the radiationresistance to the structure loss resistance. Radiated energy spreads outin space and is lost regardless of whether a receiver is present. Theenergy density of the radiated fields is a function of distance due togeometric spreading. Accordingly, the term “radiate” in all its forms asused herein refers to this form of electromagnetic propagation.

A guided electromagnetic field is a propagating electromagnetic wavewhose energy is concentrated within or near boundaries between mediahaving different electromagnetic properties. In this sense, a guidedelectromagnetic field is one that is bound to a waveguide and may becharacterized as being conveyed by the current flowing in the waveguide.If there is no load to receive and/or dissipate the energy conveyed in aguided electromagnetic wave, then no energy is lost except for thatdissipated in the conductivity of the guiding medium. Stated anotherway, if there is no load for a guided electromagnetic wave, then noenergy is consumed. Thus, a generator or other source generating aguided electromagnetic field does not deliver real power unless aresistive load is present. To this end, such a generator or other sourceessentially runs idle until a load is presented. This is akin to runninga generator to generate a 60 Hertz electromagnetic wave that istransmitted over power lines where there is no electrical load. Itshould be noted that a guided electromagnetic field or wave is theequivalent to what is termed a “transmission line mode.” This contrastswith radiated electromagnetic waves in which real power is supplied atall times in order to generate radiated waves. Unlike radiatedelectromagnetic waves, guided electromagnetic energy does not continueto propagate along a finite length waveguide after the energy source isturned off. Accordingly, the term “guide” in all its forms as usedherein refers to this transmission mode of electromagnetic propagation.

Referring now to FIG. 1, shown is a graph 100 of field strength indecibels (dB) above an arbitrary reference in volts per meter as afunction of distance in kilometers on a log-dB plot to furtherillustrate the distinction between radiated and guided electromagneticfields. The graph 100 of FIG. 1 depicts a guided field strength curve103 that shows the field strength of a guided electromagnetic field as afunction of distance. This guided field strength curve 103 isessentially the same as a transmission line mode. Also, the graph 100 ofFIG. 1 depicts a radiated field strength curve 106 that shows the fieldstrength of a radiated electromagnetic field as a function of distance.

Of interest are the shapes of the curves 103 and 106 for guided wave andfor radiation propagation, respectively. The radiated field strengthcurve 106 falls off geometrically (1/d, where d is distance), which isdepicted as a straight line on the log-log scale. The guided fieldstrength curve 103, on the other hand, has a characteristic exponentialdecay of e^(−ad)/√{square root over (d)} and exhibits a distinctive knee109 on the log-log scale. The guided field strength curve 103 and theradiated field strength curve 106 intersect at point 112, which occursat a crossing distance. At distances less than the crossing distance atintersection point 112, the field strength of a guided electromagneticfield is significantly greater at most locations than the field strengthof a radiated electromagnetic field. At distances greater than thecrossing distance, the opposite is true. Thus, the guided and radiatedfield strength curves 103 and 106 further illustrate the fundamentalpropagation difference between guided and radiated electromagneticfields. For an informal discussion of the difference between guided andradiated electromagnetic fields, reference is made to Milligan, T.,Modern Antenna Design, McGraw-Hill, 1st Edition, 1985, pp. 8-9, which isincorporated herein by reference in its entirety.

The distinction between radiated and guided electromagnetic waves, madeabove, is readily expressed formally and placed on a rigorous basis.That two such diverse solutions could emerge from one and the samelinear partial differential equation, the wave equation, analyticallyfollows from the boundary conditions imposed on the problem. The Greenfunction for the wave equation, itself, contains the distinction betweenthe nature of radiation and guided waves.

In empty space, the wave equation is a differential operator whoseeigenfunctions possess a continuous spectrum of eigenvalues on thecomplex wave-number plane. This transverse electro-magnetic (TEM) fieldis called the radiation field, and those propagating fields are called“Hertzian waves.” However, in the presence of a conducting boundary, thewave equation plus boundary conditions mathematically lead to a spectralrepresentation of wave-numbers composed of a continuous spectrum plus asum of discrete spectra. To this end, reference is made to Sommerfeld,A., “Uber die Ausbreitung der Wellen in der Drahtlosen Telegraphie,”Annalen der Physik, Vol. 28, 1909, pp. 665-736. Also see Sommerfeld, A.,“Problems of Radio,” published as Chapter 6 in Partial DifferentialEquations in Physics—Lectures on Theoretical Physics: Volume VI,Academic Press, 1949, pp. 236-289, 295-296; Collin, R. E., “HertzianDipole Radiating Over a Lossy Earth or Sea: Some Early and Late 20thCentury Controversies,” IEEE Antennas and Propagation Magazine, Vol. 46,No. 2, April 2004, pp. 64-79; and Reich, H. J., Ordnung, P. F, Krauss,H. L., and Skalnik, J. G., Microwave Theory and Techniques, VanNostrand, 1953, pp. 291-293, each of these references being incorporatedherein by reference in its entirety.

The terms “ground wave” and “surface wave” identify two distinctlydifferent physical propagation phenomena. A surface wave arisesanalytically from a distinct pole yielding a discrete component in theplane wave spectrum. See, e.g., “The Excitation of Plane Surface Waves”by Cullen, A. L., (Proceedings of the IEE (British), Vol. 101, Part IV,August 1954, pp. 225-235). In this context, a surface wave is consideredto be a guided surface wave. The surface wave (in the Zenneck-Sommerfeldguided wave sense) is, physically and mathematically, not the same asthe ground wave (in the Weyl-Norton-FCC sense) that is now so familiarfrom radio broadcasting. These two propagation mechanisms arise from theexcitation of different types of eigenvalue spectra (continuum ordiscrete) on the complex plane. The field strength of the guided surfacewave decays exponentially with distance as illustrated by curve 103 ofFIG. 1 (much like propagation in a lossy waveguide) and resemblespropagation in a radial transmission line, as opposed to the classicalHertzian radiation of the ground wave, which propagates spherically,possesses a continuum of eigenvalues, falls off geometrically asillustrated by curve 106 of FIG. 1, and results from branch-cutintegrals. As experimentally demonstrated by C. R. Burrows in “TheSurface Wave in Radio Propagation over Plane Earth” (Proceedings of theIRE, Vol. 25, No. 2, February, 1937, pp. 219-229) and “The Surface Wavein Radio Transmission” (Bell Laboratories Record, Vol. 15, June 1937,pp. 321-324), vertical antennas radiate ground waves but do not launchguided surface waves.

To summarize the above, first, the continuous part of the wave-numbereigenvalue spectrum, corresponding to branch-cut integrals, produces theradiation field, and second, the discrete spectra, and correspondingresidue sum arising from the poles enclosed by the contour ofintegration, result in non-TEM traveling surface waves that areexponentially damped in the direction transverse to the propagation.Such surface waves are guided transmission line modes. For furtherexplanation, reference is made to Friedman, B., Principles andTechniques of Applied Mathematics, Wiley, 1956, pp. pp. 214, 283-286,290, 298-300.

In free space, antennas excite the continuum eigenvalues of the waveequation, which is a radiation field, where the outwardly propagating RFenergy with ε_(z) and H_(ϕ) in-phase is lost forever. On the other hand,waveguide probes excite discrete eigenvalues, which results intransmission line propagation. See Collin, R. E., Field Theory of GuidedWaves, McGraw-Hill, 1960, pp. 453, 474-477. While such theoreticalanalyses have held out the hypothetical possibility of launching opensurface guided waves over planar or spherical surfaces of lossy,homogeneous media, for more than a century no known structures in theengineering arts have existed for accomplishing this with any practicalefficiency. Unfortunately, since it emerged in the early 1900's, thetheoretical analysis set forth above has essentially remained a theoryand there have been no known structures for practically accomplishingthe launching of open surface guided waves over planar or sphericalsurfaces of lossy, homogeneous media.

According to the various embodiments of the present disclosure, variousguided surface waveguide probes are described that are configured toexcite electric fields that couple into a guided surface waveguide modealong the surface of a lossy conducting medium. Such guidedelectromagnetic fields are substantially mode-matched in magnitude andphase to a guided surface wave mode on the surface of the lossyconducting medium. Such a guided surface wave mode can also be termed aZenneck waveguide mode. By virtue of the fact that the resultant fieldsexcited by the guided surface waveguide probes described herein aresubstantially mode-matched to a guided surface waveguide mode on thesurface of the lossy conducting medium, a guided electromagnetic fieldin the form of a guided surface wave is launched along the surface ofthe lossy conducting medium. According to one embodiment, the lossyconducting medium comprises a terrestrial medium such as the Earth.

Referring to FIG. 2, shown is a propagation interface that provides foran examination of the boundary value solutions to Maxwell's equationsderived in 1907 by Jonathan Zenneck as set forth in his paper Zenneck,J., “On the Propagation of Plane Electromagnetic Waves Along a FlatConducting Surface and their Relation to Wireless Telegraphy,” Annalender Physik, Serial 4, Vol. 23, Sep. 20, 1907, pp. 846-866. FIG. 2depicts cylindrical coordinates for radially propagating waves along theinterface between a lossy conducting medium specified as Region 1 and aninsulator specified as Region 2. Region 1 can comprise, for example, anylossy conducting medium. In one example, such a lossy conducting mediumcan comprise a terrestrial medium such as the Earth or other medium.Region 2 is a second medium that shares a boundary interface with Region1 and has different constitutive parameters relative to Region 1. Region2 can comprise, for example, any insulator such as the atmosphere orother medium. The reflection coefficient for such a boundary interfacegoes to zero only for incidence at a complex Brewster angle. SeeStratton, J. A., Electromagnetic Theory, McGraw-Hill, 1941, p. 516.

According to various embodiments, the present disclosure sets forthvarious guided surface waveguide probes that generate electromagneticfields that are substantially mode-matched to a guided surface waveguidemode on the surface of the lossy conducting medium comprising Region 1.According to various embodiments, such electromagnetic fieldssubstantially synthesize a wave front incident at a complex Brewsterangle of the lossy conducting medium that can result in zero reflection.

To explain further, in Region 2, where an e^(jωt) field variation isassumed and where ρ≠0 and z≥0 (with z being the vertical coordinatenormal to the surface of Region 1, and ρ being the radial dimension incylindrical coordinates), Zenneck's closed-form exact solution ofMaxwell's equations satisfying the boundary conditions along theinterface are expressed by the following electric field and magneticfield components:

$\begin{matrix}{{H_{2\varphi} = {{Ae}^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- j}\; {\gamma\rho}} \right)}}},} & (1) \\{{E_{2\rho} = {{A\left( \frac{u_{2}}{j\; {\omega ɛ}_{o}} \right)}e^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- j}\; {\gamma\rho}} \right)}}},{and}} & (2) \\{E_{2\; z} = {{A\left( \frac{- \gamma}{{\omega ɛ}_{o}} \right)}e^{{- u_{2}}z}{{H_{0}^{(2)}\left( {{- j}\; {\gamma\rho}} \right)}.}}} & (3)\end{matrix}$

In Region 1, where the e^(jωt) field variation is assumed and where ρ≠0and z≤0, Zenneck's closed-form exact solution of Maxwell's equationssatisfying the boundary conditions along the interface is expressed bythe following electric field and magnetic field components:

$\begin{matrix}{{H_{1\varphi} = {{Ae}^{u_{1}z}{H_{1}^{(2)}\left( {{- j}\; {\gamma\rho}} \right)}}},} & (4) \\{{E_{1\rho} = {{A\left( \frac{- u_{1}}{\sigma_{1} + {j\; {\omega ɛ}_{1}}} \right)}e^{u_{1}z}{H_{1}^{(2)}\left( {{- j}\; {\gamma\rho}} \right)}}},{and}} & (5) \\{E_{1z} = {{A\left( \frac{{- j}\; \gamma}{\sigma_{1} + {j\; {\omega ɛ}_{1}}} \right)}e^{u_{1}z}{{H_{0}^{(2)}\left( {{- j}\; {\gamma\rho}} \right)}.}}} & (6)\end{matrix}$

In these expressions, z is the vertical coordinate normal to the surfaceof Region 1 and ρ is the radial coordinate, H_(n) ⁽²⁾(−jγρ) is a complexargument Hankel function of the second kind and order n, u₁ is thepropagation constant in the positive vertical (z) direction in Region 1,u₂ is the propagation constant in the vertical (z) direction in Region2, σ₁ is the conductivity of Region 1, ω is equal to 2πf, where f is afrequency of excitation, ε₀ is the permittivity of free space, ε₁ is thepermittivity of Region 1, A is a source constant imposed by the source,and γ is a surface wave radial propagation constant.

The propagation constants in the ±z directions are determined byseparating the wave equation above and below the interface betweenRegions 1 and 2, and imposing the boundary conditions. This exercisegives, in Region 2,

$\begin{matrix}{u_{2} = \frac{- {jk}_{0}}{\sqrt{1 + \left( {ɛ_{r} - {jx}} \right)}}} & (7)\end{matrix}$

and gives, in Region 1,

u ₁ =−u ₂(ε_(r) −jx).  (8)

The radial propagation constant γ is given by

$\begin{matrix}{{\gamma = {{j\sqrt{k_{0}^{2} + u_{2}^{2}}} = {j\frac{k_{0}n}{\sqrt{1 + n^{2}}}}}},} & (9)\end{matrix}$

which is a complex expression where n is the complex index of refractiongiven by

n=√{square root over (Σ_(r) −jx)}.  (10)

In all of the above Equations,

$\begin{matrix}{{x = \frac{\sigma_{1}}{{\omega ɛ}_{o}}},{and}} & (11) \\{{k_{o} = {{\omega \sqrt{\mu_{o}ɛ_{o}}} = \frac{\lambda_{o}}{2\pi}}},} & (12)\end{matrix}$

where ε_(r) comprises the relative permittivity of Region 1, σ₁ is theconductivity of Region 1, ε₀ is the permittivity of free space, and μ₀comprises the permeability of free space. Thus, the generated surfacewave propagates parallel to the interface and exponentially decaysvertical to it. This is known as evanescence.

Thus, Equations (1)-(3) can be considered to be acylindrically-symmetric, radially-propagating waveguide mode. SeeBarlow, H. M., and Brown, J., Radio Surface Waves, Oxford UniversityPress, 1962, pp. 10-12, 29-33. The present disclosure details structuresthat excite this “open boundary” waveguide mode. Specifically, accordingto various embodiments, a guided surface waveguide probe is providedwith a charge terminal of appropriate size that is fed with voltageand/or current and is positioned relative to the boundary interfacebetween Region 2 and Region 1. This may be better understood withreference to FIG. 3, which shows an example of a guided surfacewaveguide probe 200 a that includes a charge terminal T₁ elevated abovea lossy conducting medium 203 (e.g., the Earth) along a vertical axis zthat is normal to a plane presented by the lossy conducting medium 203.The lossy conducting medium 203 makes up Region 1, and a second medium206 makes up Region 2 and shares a boundary interface with the lossyconducting medium 203.

According to one embodiment, the lossy conducting medium 203 cancomprise a terrestrial medium such as the planet Earth. To this end,such a terrestrial medium comprises all structures or formationsincluded thereon whether natural or man-made. For example, such aterrestrial medium can comprise natural elements such as rock, soil,sand, fresh water, sea water, trees, vegetation, and all other naturalelements that make up our planet. In addition, such a terrestrial mediumcan comprise man-made elements such as concrete, asphalt, buildingmaterials, and other man-made materials. In other embodiments, the lossyconducting medium 203 can comprise some medium other than the Earth,whether naturally occurring or man-made. In other embodiments, the lossyconducting medium 203 can comprise other media such as man-made surfacesand structures such as automobiles, aircraft, man-made materials (suchas plywood, plastic sheeting, or other materials) or other media.

In the case where the lossy conducting medium 203 comprises aterrestrial medium or Earth, the second medium 206 can comprise theatmosphere above the ground. As such, the atmosphere can be termed an“atmospheric medium” that comprises air and other elements that make upthe atmosphere of the Earth. In addition, it is possible that the secondmedium 206 can comprise other media relative to the lossy conductingmedium 203.

The guided surface waveguide probe 200 a includes a feed network 209that couples an excitation source 212 to the charge terminal T₁ via,e.g., a vertical feed line conductor and/or other components as will bedescribed. According to various embodiments, a charge Q₁ is imposed onthe charge terminal T₁ to synthesize an electric field based upon thevoltage applied to terminal T₁ at any given instant. Depending on theangle of incidence (θ_(i)) of the electric field (E), it is possible tosubstantially mode-match the electric field to a guided surfacewaveguide mode on the surface of the lossy conducting medium 203comprising Region 1.

By considering the Zenneck closed-form solutions of Equations (1)-(6),the Leontovich impedance boundary condition between Region 1 and Region2 can be stated as

{circumflex over (z)}×{right arrow over (H)} ₂(ρ,φ,0)={right arrow over(J)} _(S),  (13)

where {circumflex over (z)} is a unit normal in the positive vertical(+z) direction and {right arrow over (H)}₂ is the magnetic fieldstrength in Region 2 expressed by Equation (1) above. Equation (13)implies that the electric and magnetic fields specified in Equations(1)-(3) may result in a radial surface current density along theboundary interface, where the radial surface current density can bespecified by

J _(ρ)(ρ′)=−AH ₁ ⁽²⁾(−jγρ′)  (14)

where A is a constant. Further, it should be noted that close-in to theguided surface waveguide probe 200 (for ρ<<λ), Equation (14) above hasthe behavior

$\begin{matrix}{{J_{close}\left( \rho^{\prime} \right)} = {\frac{- {A\left( {j\; 2} \right)}}{\pi \left( {{- j}\; {\gamma\rho}^{\prime}} \right)} = {{- H_{\varphi}} = {- {\frac{I_{o}}{2\pi \; \rho^{\prime}}.}}}}} & (15)\end{matrix}$

The negative sign means that when source current (I₀) flows verticallyupward as illustrated in FIG. 3, the “close-in” ground current flowsradially inward. By field matching on H_(ϕ)“close-in,” it can bedetermined that

$\begin{matrix}{A = {{- \frac{I_{o}\gamma}{4}} = {- \frac{\omega \; q_{1}\gamma}{4}}}} & (16)\end{matrix}$

where q₁=C₁V₁, in Equations (1)-(6) and (14). Therefore, the radialsurface current density of Equation (14) can be restated as

$\begin{matrix}{{J_{\rho}\left( \rho^{\prime} \right)} = {\frac{I_{o}\gamma}{4}{{H_{1}^{(2)}\left( {{- j}\; \gamma \; \rho^{\prime}} \right)}.}}} & (17)\end{matrix}$

The fields expressed by Equations (1)-(6) and (17) have the nature of atransmission line mode bound to a lossy interface, not radiation fieldsthat are associated with groundwave propagation. See Barlow, H. M. andBrown, J., Radio Surface Waves, Oxford University Press, 1962, pp. 1-5.

At this point, a review of the nature of the Hankel functions used inEquations (1)-(6) and (17) is provided for these solutions of the waveequation. One might observe that the Hankel functions of the first andsecond kind and order n are defined as complex combinations of thestandard Bessel functions of the first and second kinds

H _(n) ⁽¹⁾(x)=J _(n)(x)+jN _(n)(x), and  (18)

H _(n) ⁽²⁾(x)=J _(n)(x)−JN _(n)(x),  (19)

These functions represent cylindrical waves propagating radially inward(H_(n) ⁽¹⁾) and outward (H_(n) ⁽²⁾), respectively. The definition isanalogous to the relationship e^(±jx)=cos x±j sin x. See, for example,Harrington, R. F., Time-Harmonic Fields, McGraw-Hill, 1961, pp. 460-463.

That H_(n) ⁽²⁾(k_(ρ)ρ) is an outgoing wave can be recognized from itslarge argument asymptotic behavior that is obtained directly from theseries definitions of J_(n)(x) and N_(n)(x). Far-out from the guidedsurface waveguide probe:

$\begin{matrix}{{{{H_{n}^{(2)}(x)}\underset{x\rightarrow\infty}{\rightarrow}{\sqrt{\frac{2j}{\pi \; x}}j^{n}e^{- {jx}}}} = {\sqrt{\frac{2}{\pi \; x}}j^{n}e^{- {j{({x - \frac{\pi}{4}})}}}}},} & \left( {20a} \right)\end{matrix}$

which, when multiplied by e^(jωt), is an outward propagating cylindricalwave of the form e^(j(ωt−kρ)) with a 1/√{square root over (ρ)}spatialvariation. The first order (n=1) solution can be determined fromEquation (20a) to be

$\begin{matrix}{{{H_{1}^{(2)}(x)}\underset{x\rightarrow\infty}{\rightarrow}{j\sqrt{\frac{2j}{\pi \; x}}e^{- {jx}}}} = {\sqrt{\frac{2}{\pi \; x}}{e^{- {j{({x - \frac{\pi}{2} - \frac{\pi}{4}})}}}.}}} & \left( {20b} \right)\end{matrix}$

Close-in to the guided surface waveguide probe (for ρ<<λ), the Hankelfunction of first order and the second kind behaves as

$\begin{matrix}{{H_{1}^{(2)}(x)}\underset{x\rightarrow 0}{\rightarrow}\; {\frac{2j}{\pi \; x}.}} & (21)\end{matrix}$

Note that these asymptotic expressions are complex quantities. When x isa real quantity, Equations (20b) and (21) differ in phase by √{squareroot over (j)}, which corresponds to an extra phase advance or “phaseboost” of 45° or, equivalently, λ/8. The close-in and far-out asymptotesof the first order Hankel function of the second kind have a Hankel“crossover” or transition point where they are of equal magnitude at adistance of ρ=R_(x).

Thus, beyond the Hankel crossover point the “far out” representationpredominates over the “close-in” representation of the Hankel function.The distance to the Hankel crossover point (or Hankel crossoverdistance) can be found by equating Equations (20b) and (21) for −jγρ,and solving for R_(x). With x=σ/ωε₀, it can be seen that the far-out andclose-in Hankel function asymptotes are frequency dependent, with theHankel crossover point moving out as the frequency is lowered. It shouldalso be noted that the Hankel function asymptotes may also vary as theconductivity (σ) of the lossy conducting medium changes. For example,the conductivity of the soil can vary with changes in weatherconditions.

Referring to FIG. 4, shown is an example of a plot of the magnitudes ofthe first order Hankel functions of Equations (20b) and (21) for aRegion 1 conductivity of σ=0.010 mhos/m and relative permittivityε_(r)=15, at an operating frequency of 1850 kHz. Curve 115 is themagnitude of the far-out asymptote of Equation (20b) and curve 118 isthe magnitude of the close-in asymptote of Equation (21), with theHankel crossover point 121 occurring at a distance of R_(x)=54 feet.While the magnitudes are equal, a phase offset exists between the twoasymptotes at the Hankel crossover point 121. It can also be seen thatthe Hankel crossover distance is much less than a wavelength of theoperation frequency.

Considering the electric field components given by Equations (2) and (3)of the Zenneck closed-form solution in Region 2, it can be seen that theratio of ε_(z) and ε_(ρ) asymptotically passes to

$\begin{matrix}{{\frac{E_{z}}{E_{\rho}} = {{{\left( \frac{{- j}\; \gamma}{u_{2}} \right)\frac{H_{0}^{(2)}\left( {{- j}\; \gamma \; \rho} \right)}{H_{1}^{(2)}\left( {{- j}\; \gamma \; \rho} \right)}}\underset{\rho\rightarrow\infty}{\rightarrow}\sqrt{ɛ_{r} - {j\frac{\sigma}{\omega \; ɛ_{o}}}}} = {n = {\tan \; \theta_{i}}}}},} & (22)\end{matrix}$

where n is the complex index of refraction of Equation (10) and θ_(i) isthe angle of incidence of the electric field. In addition, the verticalcomponent of the mode-matched electric field of Equation (3)asymptotically passes to

$\begin{matrix}{{{E_{2z}\underset{\rho->\infty}{}\left( \frac{q_{free}}{ɛ_{o}} \right)}\sqrt{\frac{\gamma^{3}}{8\pi}}e^{{- u_{2}}z}\frac{e^{- {j{({{\gamma \; \rho} - {\pi/4}})}}}}{\sqrt{\rho}}},} & (23)\end{matrix}$

which is linearly proportional to free charge on the isolated componentof the elevated charge terminal's capacitance at the terminal voltage,q_(free)=C_(free)×V_(T).

For example, the height H₁ of the elevated charge terminal T₁ in FIG. 3affects the amount of free charge on the charge terminal T₁. When thecharge terminal T₁ is near the ground plane of Region 1, most of thecharge Q₁ on the terminal is “bound.” As the charge terminal T₁ iselevated, the bound charge is lessened until the charge terminal T₁reaches a height at which substantially all of the isolated charge isfree.

The advantage of an increased capacitive elevation for the chargeterminal T₁ is that the charge on the elevated charge terminal T₁ isfurther removed from the ground plane, resulting in an increased amountof free charge q_(free) to couple energy into the guided surfacewaveguide mode. As the charge terminal T₁ is moved away from the groundplane, the charge distribution becomes more uniformly distributed aboutthe surface of the terminal. The amount of free charge is related to theself-capacitance of the charge terminal T₁.

For example, the capacitance of a spherical terminal can be expressed asa function of physical height above the ground plane. The capacitance ofa sphere at a physical height of h above a perfect ground is given by

C _(elevated sphere)=4πε₀ a(1+M+M ² +M ³+2M ⁴+3M ⁵+ . . . ),  (24)

where the diameter of the sphere is 2a, and where M=a/2h with h beingthe height of the spherical terminal. As can be seen, an increase in theterminal height h reduces the capacitance C of the charge terminal. Itcan be shown that for elevations of the charge terminal T₁ that are at aheight of about four times the diameter (4D=8a) or greater, the chargedistribution is approximately uniform about the spherical terminal,which can improve the coupling into the guided surface waveguide mode.

In the case of a sufficiently isolated terminal, the self-capacitance ofa conductive sphere can be approximated by C=4πε₀a, where a is theradius of the sphere in meters, and the self-capacitance of a disk canbe approximated by C=8ε₀a, where a is the radius of the disk in meters.The charge terminal T₁ can include any shape such as a sphere, a disk, acylinder, a cone, a torus, a hood, one or more rings, or any otherrandomized shape or combination of shapes. An equivalent sphericaldiameter can be determined and used for positioning of the chargeterminal T₁.

This may be further understood with reference to the example of FIG. 3,where the charge terminal T₁ is elevated at a physical height ofh_(p)=H₁ above the lossy conducting medium 203. To reduce the effects ofthe “bound” charge, the charge terminal T₁ can be positioned at aphysical height that is at least four times the spherical diameter (orequivalent spherical diameter) of the charge terminal T₁ to reduce thebounded charge effects.

Referring next to FIG. 5A, shown is a ray optics interpretation of theelectric field produced by the elevated charge Q₁ on charge terminal T₁of FIG. 3. As in optics, minimizing the reflection of the incidentelectric field can improve and/or maximize the energy coupled into theguided surface waveguide mode of the lossy conducting medium 203. For anelectric field (E_(∥)) that is polarized parallel to the plane ofincidence (not the boundary interface), the amount of reflection of theincident electric field may be determined using the Fresnel reflectioncoefficient, which can be expressed as

$\begin{matrix}{{{\Gamma_{||}\left( \theta_{i} \right)} = {\frac{E_{||{,R}}}{E_{||{,i}}} = \frac{\sqrt{\left( {ɛ_{r} - {jx}} \right) - {\sin^{2}\mspace{14mu} \theta_{i}}} - {\left( {ɛ_{r} - {jx}} \right)\mspace{14mu} \cos \mspace{14mu} \theta_{i}}}{\sqrt{\left( {ɛ_{r} - {jx}} \right) - {\sin^{2}\mspace{14mu} \theta_{i}}} + {\left( {ɛ_{r} - {jx}} \right)\mspace{14mu} \cos \mspace{14mu} \theta_{i}}}}},} & (25)\end{matrix}$

where θ_(i) is the conventional angle of incidence measured with respectto the surface normal.

In the example of FIG. 5A, the ray optic interpretation shows theincident field polarized parallel to the plane of incidence having anangle of incidence of θ_(i), which is measured with respect to thesurface normal ({circumflex over (z)}). There will be no reflection ofthe incident electric field when Γ_(∥)(θ_(i))=0 and thus the incidentelectric field will be completely coupled into a guided surfacewaveguide mode along the surface of the lossy conducting medium 203. Itcan be seen that the numerator of Equation (25) goes to zero when theangle of incidence is

θ_(i)=arctan({right arrow over (ε_(r) −jx)})=θ_(i,B),  (26)

where x=σ/ωε₀. This complex angle of incidence (θ_(i,B)) is referred toas the Brewster angle. Referring back to Equation (22), it can be seenthat the same complex Brewster angle (θ_(i,B)) relationship is presentin both Equations (22) and (26).

As illustrated in FIG. 5A, the electric field vector E can be depictedas an incoming non-uniform plane wave, polarized parallel to the planeof incidence. The electric field vector E can be created fromindependent horizontal and vertical components as

{right arrow over (E)}(θ_(i))=E _(ρ) {circumflex over (ρ)}+E _(z){circumflex over (z)}.  (27)

Geometrically, the illustration in FIG. 5A suggests that the electricfield vector E can be given by

$\begin{matrix}{{{E_{\rho}\left( {\rho,z} \right)} = {{E\left( {\rho,z} \right)}\mspace{14mu} \cos \mspace{14mu} \theta_{i}}},{and}} & \left( {28a} \right) \\{{{E_{z}\left( {\rho,z} \right)} = {{{E\left( {\rho,z} \right)}\mspace{14mu} {\cos \left( {\frac{\pi}{2} - \theta_{i}} \right)}} = {{E\left( {\rho,z} \right)}\mspace{14mu} \sin \mspace{14mu} \theta_{i}}}},} & \left( {28b} \right)\end{matrix}$

which means that the field ratio is

$\begin{matrix}{\frac{E_{\rho}}{E_{z}} = {\frac{1}{\tan \mspace{14mu} \theta_{i}} = {\tan \mspace{14mu} {\psi_{i}.}}}} & (29)\end{matrix}$

A generalized parameter W, called “wave tilt,” is noted herein as theratio of the horizontal electric field component to the verticalelectric field component given by

$\begin{matrix}{{W = {\frac{E_{\rho}}{E_{z}} = \left| W \middle| e^{j\; \Psi} \right.}},{or}} & \left( {30a} \right) \\{{\frac{1}{W} = {\frac{E_{z}}{E_{\rho}} = {{\tan \mspace{14mu} \theta_{i}} = {\frac{1}{|W|}e^{{- j}\; \Psi}}}}},} & \left( {30b} \right)\end{matrix}$

which is complex and has both magnitude and phase. For anelectromagnetic wave in Region 2, the wave tilt angle (Ψ) is equal tothe angle between the normal of the wave-front at the boundary interfacewith Region 1 and the tangent to the boundary interface. This may beeasier to see in FIG. 5B, which illustrates equi-phase surfaces of anelectromagnetic wave and their normals for a radial cylindrical guidedsurface wave. At the boundary interface (z=0) with a perfect conductor,the wave-front normal is parallel to the tangent of the boundaryinterface, resulting in W=0. However, in the case of a lossy dielectric,a wave tilt W exists because the wave-front normal is not parallel withthe tangent of the boundary interface at z=0.

Applying Equation (30b) to a guided surface wave gives

$\begin{matrix}{{\tan \mspace{14mu} \theta_{i,B}} = {\frac{E_{z}}{E_{\rho}} = {\frac{u_{2}}{\gamma} = {\sqrt{ɛ_{r} - {jx}} = {n = {\frac{1}{W} = {\frac{1}{|W|}{e^{{- j}\; \Psi}.}}}}}}}} & (31)\end{matrix}$

With the angle of incidence equal to the complex Brewster angle(θ_(i,B)), the Fresnel reflection coefficient of Equation (25) vanishes,as shown by

$\begin{matrix}{{\Gamma_{||}\left( \theta_{i,B} \right)} = {\left. \frac{\sqrt{\left( {ɛ_{r} - {jx}} \right) - {\sin^{2}\mspace{14mu} \theta_{i}}} - {\left( {ɛ_{r} - {jx}} \right)\mspace{14mu} \cos \mspace{14mu} \theta_{i}}}{\sqrt{\left( {ɛ_{r} - {jx}} \right) - {\sin^{2}\mspace{14mu} \theta_{i}}} + {\left( {ɛ_{r} - {jx}} \right)\mspace{14mu} \cos \mspace{14mu} \theta_{i}}} \right|_{\theta_{i} = \theta_{i,B}} = 0.}} & (32)\end{matrix}$

By adjusting the complex field ratio of Equation (22), an incident fieldcan be synthesized to be incident at a complex angle at which thereflection is reduced or eliminated. Establishing this ratio asn=√{square root over (ε_(r)−jx)} results in the synthesized electricfield being incident at the complex Brewster angle, making thereflections vanish.

The concept of an electrical effective height can provide furtherinsight into synthesizing an electric field with a complex angle ofincidence with a guided surface waveguide probe 200. The electricaleffective height (h_(eff)) has been defined as

$\begin{matrix}{h_{eff} = {\frac{1}{I_{0}}{\int_{0}^{h_{p}}{{I(z)}{dz}}}}} & (33)\end{matrix}$

for a monopole with a physical height (or length) of h_(p). Since theexpression depends upon the magnitude and phase of the sourcedistribution along the structure, the effective height (or length) iscomplex in general. The integration of the distributed current I(z) ofthe structure is performed over the physical height of the structure(h_(p)), and normalized to the ground current (I₀) flowing upwardthrough the base (or input) of the structure. The distributed currentalong the structure can be expressed by

I(z)=I _(C) cos(β₀ z),  (34)

where β₀ is the propagation factor for current propagating on thestructure. In the example of FIG. 3, I_(C) is the current that isdistributed along the vertical structure of the guided surface waveguideprobe 200 a.

For example, consider a feed network 209 that includes a low loss coil(e.g., a helical coil) at the bottom of the structure and a verticalfeed line conductor connected between the coil and the charge terminalT₁. The phase delay due to the coil (or helical delay line) isθ_(c)=β_(p)l_(C), with a physical length of l_(C) and a propagationfactor of

$\begin{matrix}{{\beta_{p} = {\frac{2\pi}{\lambda_{p}} = \frac{2\pi}{V_{f}\lambda_{0}}}},} & (35)\end{matrix}$

where V_(f) is the velocity factor on the structure, λ₀ is thewavelength at the supplied frequency, and λ_(p) is the propagationwavelength resulting from the velocity factor V_(f). The phase delay ismeasured relative to the ground (stake) current I₀.

In addition, the spatial phase delay along the length l_(w) of thevertical feed line conductor can be given by θ_(y)=β_(w)l_(w) whereβ_(w) is the propagation phase constant for the vertical feed lineconductor. In some implementations, the spatial phase delay may beapproximated by θ_(y)=β_(w)h_(p), since the difference between thephysical height h_(p) of the guided surface waveguide probe 200 a andthe vertical feed line conductor length l_(w) is much less than awavelength at the supplied frequency (λ₀). As a result, the total phasedelay through the coil and vertical feed line conductor isΦ=θ_(c)+θ_(y), and the current fed to the top of the coil from thebottom of the physical structure is

I _(C)(θ_(c)+θ_(y))=I ₀ e ^(jΦ),  (36)

with the total phase delay Φ measured relative to the ground (stake)current I₀. Consequently, the electrical effective height of a guidedsurface waveguide probe 200 can be approximated by

$\begin{matrix}{{h_{eff} = {{\frac{1}{I_{0}}{\int_{0}^{h_{p}}{I_{0}e^{j\; \Phi}{\cos \left( {\beta_{0}z} \right)}\ {dz}}}} \cong {h_{p}e^{j\; \Phi}}}},} & (37)\end{matrix}$

for the case where the physical height h_(p)<<λ₀. The complex effectiveheight of a monopole, h_(eff)=h_(p) at an angle (or phase shift) of Φ,may be adjusted to cause the source fields to match a guided surfacewaveguide mode and cause a guided surface wave to be launched on thelossy conducting medium 203.

In the example of FIG. 5A, ray optics are used to illustrate the complexangle trigonometry of the incident electric field (E) having a complexBrewster angle of incidence (θ_(i,B)) at the Hankel crossover distance(R_(x)) 121. Recall from Equation (26) that, for a lossy conductingmedium, the Brewster angle is complex and specified by

$\begin{matrix}{{\tan \mspace{14mu} \theta_{i,B}} = {\sqrt{ɛ_{r} - {j\frac{\sigma}{{\omega ɛ}_{o}}}} = {n.}}} & (38)\end{matrix}$

Electrically, the geometric parameters are related by the electricaleffective height (h_(eff)) of the charge terminal T₁ by

R _(x) tan ψ_(i,B) =R _(x) ×W=h _(eff) =h _(p) e ^(jΦ),  (39)

where ψ_(i,B)=(π/2)−θ_(i,B) is the Brewster angle measured from thesurface of the lossy conducting medium. To couple into the guidedsurface waveguide mode, the wave tilt of the electric field at theHankel crossover distance can be expressed as the ratio of theelectrical effective height and the Hankel crossover distance

$\begin{matrix}{\frac{h_{eff}}{R_{x}} = {{\tan \; \psi_{i,B}} = {W_{Rx}.}}} & (40)\end{matrix}$

Since both the physical height (h_(p)) and the Hankel crossover distance(R_(x)) are real quantities, the angle (Ψ) of the desired guided surfacewave tilt at the Hankel crossover distance (R_(x)) is equal to the phase(Φ) of the complex effective height (h_(eff)). This implies that byvarying the phase at the supply point of the coil, and thus the phaseshift in Equation (37), the phase, Φ, of the complex effective heightcan be manipulated to match the angle of the wave tilt, Ψ, of the guidedsurface waveguide mode at the Hankel crossover point 121: Φ=Ψ.

In FIG. 5A, a right triangle is depicted having an adjacent side oflength R_(x) along the lossy conducting medium surface and a complexBrewster angle ψ_(i,B) measured between a ray 124 extending between theHankel crossover point 121 at R_(x) and the center of the chargeterminal T₁, and the lossy conducting medium surface 127 between theHankel crossover point 121 and the charge terminal T₁. With the chargeterminal T₁ positioned at physical height h_(p) and excited with acharge having the appropriate phase delay Φ, the resulting electricfield is incident with the lossy conducting medium boundary interface atthe Hankel crossover distance R_(x), and at the Brewster angle. Underthese conditions, the guided surface waveguide mode can be excitedwithout reflection or substantially negligible reflection.

If the physical height of the charge terminal T₁ is decreased withoutchanging the phase shift Φ of the effective height (h_(eff)), theresulting electric field intersects the lossy conducting medium 203 atthe Brewster angle at a reduced distance from the guided surfacewaveguide probe 200. FIG. 6 graphically illustrates the effect ofdecreasing the physical height of the charge terminal T₁ on the distancewhere the electric field is incident at the Brewster angle. As theheight is decreased from h₃ through h₂ to h₁, the point where theelectric field intersects with the lossy conducting medium (e.g., theEarth) at the Brewster angle moves closer to the charge terminalposition. However, as Equation (39) indicates, the height H₁ (FIG. 3) ofthe charge terminal T₁ should be at or higher than the physical height(h_(p)) in order to excite the far-out component of the Hankel function.With the charge terminal T₁ positioned at or above the effective height(h_(eff)), the lossy conducting medium 203 can be illuminated at theBrewster angle of incidence (ψ_(i,B)=(π/2)−θ_(i,B)) at or beyond theHankel crossover distance (R_(x)) 121 as illustrated in FIG. 5A. Toreduce or minimize the bound charge on the charge terminal T₁, theheight should be at least four times the spherical diameter (orequivalent spherical diameter) of the charge terminal T₁ as mentionedabove.

A guided surface waveguide probe 200 can be configured to establish anelectric field having a wave tilt that corresponds to a waveilluminating the surface of the lossy conducting medium 203 at a complexBrewster angle, thereby exciting radial surface currents bysubstantially mode-matching to a guided surface wave mode at (or beyond)the Hankel crossover point 121 at R_(x).

Referring to FIG. 7, shown is a graphical representation of an exampleof a guided surface waveguide probe 200 b that includes a chargeterminal T₁. An AC source 212 acts as the excitation source for thecharge terminal T₁, which is coupled to the guided surface waveguideprobe 200 b through a feed network 209 (FIG. 3) comprising a coil 215such as, e.g., a helical coil. In other implementations, the AC source212 can be inductively coupled to the coil 215 through a primary coil.In some embodiments, an impedance matching network may be included toimprove and/or maximize coupling of the AC source 212 to the coil 215.

As shown in FIG. 7, the guided surface waveguide probe 200 b can includethe upper charge terminal T₁ (e.g., a sphere at height h_(p)) that ispositioned along a vertical axis z that is substantially normal to theplane presented by the lossy conducting medium 203. A second medium 206is located above the lossy conducting medium 203. The charge terminal T₁has a self-capacitance C_(T). During operation, charge Q₁ is imposed onthe terminal T₁ depending on the voltage applied to the terminal T₁ atany given instant.

In the example of FIG. 7, the coil 215 is coupled to a ground stake 218at a first end and to the charge terminal T₁ via a vertical feed lineconductor 221. In some implementations, the coil connection to thecharge terminal T₁ can be adjusted using a tap 224 of the coil 215 asshown in FIG. 7. The coil 215 can be energized at an operating frequencyby the AC source 212 through a tap 227 at a lower portion of the coil215. In other implementations, the AC source 212 can be inductivelycoupled to the coil 215 through a primary coil.

The construction and adjustment of the guided surface waveguide probe200 is based upon various operating conditions, such as the transmissionfrequency, conditions of the lossy conducting medium (e.g., soilconductivity σ and relative permittivity ε_(r)), and size of the chargeterminal T₁. The index of refraction can be calculated from Equations(10) and (11) as

n=√{square root over (ε_(r) −jx)},  (41)

where x=σ/ωε₀ with ω=2πf. The conductivity a and relative permittivityε_(r) can be determined through test measurements of the lossyconducting medium 203. The complex Brewster angle (θ_(i,B)) measuredfrom the surface normal can also be determined from Equation (26) as

θ_(i,B)=arctan(√{square root over (ε_(r) −jx)}),  (42)

or measured from the surface as shown in FIG. 5A as

$\begin{matrix}{\psi_{i,B} = {\frac{\pi}{2} - {\theta_{i,B}.}}} & (43)\end{matrix}$

The wave tilt at the Hankel crossover distance (W_(Rx)) can also befound using Equation (40).

The Hankel crossover distance can also be found by equating themagnitudes of Equations (20b) and (21) for −jγρ, and solving for R_(x)as illustrated by FIG. 4. The electrical effective height can then bedetermined from Equation (39) using the Hankel crossover distance andthe complex Brewster angle as

h _(eff) =h _(p) e ^(jΦ) =R _(x) tan ψ_(i,B).  (44)

As can be seen from Equation (44), the complex effective height(h_(eff)) includes a magnitude that is associated with the physicalheight (h_(p)) of the charge terminal T₁ and a phase delay (Φ) that isto be associated with the angle (Ψ) of the wave tilt at the Hankelcrossover distance (R_(x)). With these variables and the selected chargeterminal T₁ configuration, it is possible to determine the configurationof a guided surface waveguide probe 200.

With the charge terminal T₁ positioned at or above the physical height(h_(p)), the feed network 209 (FIG. 3) and/or the vertical feed lineconnecting the feed network to the charge terminal T₁ can be adjusted tomatch the phase (Φ) of the charge Q₁ on the charge terminal T₁ to theangle (Ψ) of the wave tilt (W). The size of the charge terminal T₁ canbe chosen to provide a sufficiently large surface for the charge Q₁imposed on the terminals. In general, it is desirable to make the chargeterminal T₁ as large as practical. The size of the charge terminal T₁should be large enough to avoid ionization of the surrounding air, whichcan result in electrical discharge or sparking around the chargeterminal.

The phase delay θ_(c) of a helically-wound coil can be determined fromMaxwell's equations as has been discussed by Corum, K. L. and J. F.Corum, “RF Coils, Helical Resonators and Voltage Magnification byCoherent Spatial Modes,” Microwave Review, Vol. 7, No. 2, September2001, pp. 36-45., which is incorporated herein by reference in itsentirety. For a helical coil with H/D>1, the ratio of the velocity ofpropagation (ν) of a wave along the coil's longitudinal axis to thespeed of light (c), or the “velocity factor,” is given by

$\begin{matrix}{{V_{f} = {\frac{\upsilon}{c} = \frac{1}{\sqrt{1 + {20\left( \frac{D}{s} \right)^{2.5}\left( \frac{D}{\lambda_{o}} \right)^{0.5}}}}}},} & (45)\end{matrix}$

where H is the axial length of the solenoidal helix, D is the coildiameter, N is the number of turns of the coil, s=H/N is theturn-to-turn spacing (or helix pitch) of the coil, and λ₀ is thefree-space wavelength. Based upon this relationship, the electricallength, or phase delay, of the helical coil is given by

$\begin{matrix}{\theta_{c} = {{\beta_{p}H} = {{\frac{2\pi}{\lambda_{p}}H} = {\frac{2\pi}{V_{f}\lambda_{0}}{H.}}}}} & (46)\end{matrix}$

The principle is the same if the helix is wound spirally or is short andfat, but V_(f) and θ_(c) are easier to obtain by experimentalmeasurement. The expression for the characteristic (wave) impedance of ahelical transmission line has also been derived as

$\begin{matrix}{Z_{c} = {{\frac{60}{V_{f}}\left\lbrack {{\ln \left( \frac{V_{f}\lambda_{0}}{D} \right)} - 1.027} \right\rbrack}.}} & (47)\end{matrix}$

The spatial phase delay θ_(y) of the structure can be determined usingthe traveling wave phase delay of the vertical feed line conductor 221(FIG. 7). The capacitance of a cylindrical vertical conductor above aperfect ground plane can be expressed as

$\begin{matrix}{{C_{A} = {\frac{2{\pi ɛ}_{o}h_{w}}{{\ln \left( \frac{h}{a} \right)} - 1}{Farads}}},} & (48)\end{matrix}$

where h_(w) is the vertical length (or height) of the conductor and a isthe radius (in mks units). As with the helical coil, the traveling wavephase delay of the vertical feed line conductor can be given by

$\begin{matrix}{{\theta_{y} = {{\beta_{w}h_{w}} = {{\frac{2\pi}{\lambda_{w}}h_{w}} = {\frac{2\pi}{V_{w}\lambda_{0}}h_{w}}}}},} & (49)\end{matrix}$

where β_(w) is the propagation phase constant for the vertical feed lineconductor, h_(w) is the vertical length (or height) of the vertical feedline conductor, V_(w) is the velocity factor on the wire, λ₀ is thewavelength at the supplied frequency, and λ_(w) is the propagationwavelength resulting from the velocity factor V_(w). For a uniformcylindrical conductor, the velocity factor is a constant withV_(w)≈0.94, or in a range from about 0.93 to about 0.98. If the mast isconsidered to be a uniform transmission line, its average characteristicimpedance can be approximated by

$\begin{matrix}{{Z_{w} = {\frac{60}{V_{w}}\left\lbrack {{\ln \left( \frac{h_{w}}{a} \right)} - 1} \right\rbrack}},} & (50)\end{matrix}$

where V_(w)≈0.94 for a uniform cylindrical conductor and a is the radiusof the conductor. An alternative expression that has been employed inamateur radio literature for the characteristic impedance of asingle-wire feed line can be given by

$\begin{matrix}{Z_{w} = {138\mspace{14mu} {{\log \left( \frac{1.123\mspace{14mu} V_{w}\lambda_{0}}{2\pi \; a} \right)}.}}} & (51)\end{matrix}$

Equation (51) implies that Z_(w) for a single-wire feeder varies withfrequency. The phase delay can be determined based upon the capacitanceand characteristic impedance.

The coil 215 is one example of a phase delay circuit that may beemployed as part of the feed network 209. It is understood that othercomponents may be used in the place of the coil 215 as a phase delaycircuit in a given embodiment. Such alternative phase delay circuitscomprise circuits that provide for a transmission line delay. Suchelements provide for the phase delay associated with transmission lines.Accordingly, other examples of phase delay circuits may include sectionsof transmission lines, circuits that mimic transmission lines such asL-C circuits, sections of coaxial cable or other cables, active circuitsthat employ operational amplifiers and the like, and other phase delaycircuits. It should be noted that whenever an embodiment of a guidedsurface waveguide probe is discussed herein that includes the use of acoil as phase delay circuit, it is understood that the coil is cited asan example of a phase delay circuit and that other types of phase delaycircuits may also be used.

With a charge terminal T₁ positioned over the lossy conducting medium203 as shown in FIG. 3, the feed network 209 can be adjusted to excitethe charge terminal T₁ with the phase shift (Φ) of the complex effectiveheight (h_(eff)) equal to the angle (Ψ) of the wave tilt at the Hankelcrossover distance, or Φ=Ψ. When this condition is met, the electricfield produced by the charge oscillating Q₁ on the charge terminal T₁ iscoupled into a guided surface waveguide mode traveling along the surfaceof a lossy conducting medium 203. For example, if the Brewster angle(θ_(i,B)), the phase delay (θ_(y)) associated with the vertical feedline conductor 221 (FIG. 7), and the configuration of the coil 215 (FIG.7) are known, then the position of the tap 224 (FIG. 7) can bedetermined and adjusted to impose an oscillating charge Q₁ on the chargeterminal T₁ with phase Φ=Ψ. The position of the tap 224 may be adjustedto maximize coupling the traveling surface waves into the guided surfacewaveguide mode. Excess coil length beyond the position of the tap 224can be removed to reduce the capacitive effects. The vertical wireheight and/or the geometrical parameters of the helical coil may also bevaried.

The coupling to the guided surface waveguide mode on the surface of thelossy conducting medium 203 can be improved and/or optimized by tuningthe guided surface waveguide probe 200 for standing wave resonance withrespect to a complex image plane associated with the charge Q₁ on thecharge terminal T₁. By doing this, the performance of the guided surfacewaveguide probe 200 can be adjusted for increased and/or maximum voltage(and thus charge Q₁) on the charge terminal T₁. Referring back to FIG.3, the effect of the lossy conducting medium 203 in Region 1 can beexamined using image theory analysis.

Physically, an elevated charge Q₁ placed over a perfectly conductingplane attracts the free charge on the perfectly conducting plane, whichthen “piles up” in the region under the elevated charge Q₁. Theresulting distribution of “bound” electricity on the perfectlyconducting plane is similar to a bell-shaped curve. The superposition ofthe potential of the elevated charge Q₁, plus the potential of theinduced “piled up” charge beneath it, forces a zero equipotentialsurface for the perfectly conducting plane. The boundary value problemsolution that describes the fields in the region above the perfectlyconducting plane may be obtained using the classical notion of imagecharges, where the field from the elevated charge is superimposed withthe field from a corresponding “image” charge below the perfectlyconducting plane.

This analysis may also be used with respect to a lossy conducting medium203 by assuming the presence of an effective image charge Q₁′ beneaththe guided surface waveguide probe 200. The effective image charge Q₁′coincides with the charge Q₁ on the charge terminal T₁ about aconducting image ground plane 130, as illustrated in FIG. 3. However,the image charge Q₁′ is not merely located at some real depth and 180°out of phase with the primary source charge Q₁ on the charge terminal asthey would be in the case of a perfect conductor. Rather, the lossyconducting medium 203 (e.g., a terrestrial medium) presents a phaseshifted image. That is to say, the image charge Q₁′ is at a complexdepth below the surface (or physical boundary) of the lossy conductingmedium 203. For a discussion of complex image depth, reference is madeto Wait, J. R., “Complex Image Theory—Revisited,” IEEE Antennas andPropagation Magazine, Vol. 33, No. 4, August 1991, pp. 27-29, which isincorporated herein by reference in its entirety.

Instead of the image charge Q₁′ being at a depth that is equal to thephysical height (H₁) of the charge Q₁, the conducting image ground plane130 (representing a perfect conductor) is located at a complex depth ofz=−d/2 and the image charge Q₁′ appears at a complex depth (i.e., the“depth” has both magnitude and phase), given by −D₁=−(d/2+d/2+H₁)≠H₁.For vertically polarized sources over the Earth,

$\begin{matrix}{{d = {{\frac{\sqrt[2]{\gamma_{e}^{2} + k_{0}^{2}}}{\gamma_{e}^{2}} \approx \frac{2}{\gamma_{e}}} = {{d_{r} + {jd}_{i}} = \left| d \middle| {\angle\zeta} \right.}}},} & (52)\end{matrix}$

where

γ_(e) ² =jωμ ₁σ₁−ω²μ₁ε₁, and  (53)

k ₀=ω√{square root over (μ₀ε₀)},  (54)

as indicated in Equation (12). The complex spacing of the image charge,in turn, implies that the external field will experience extra phaseshifts not encountered when the interface is either a dielectric or aperfect conductor. In the lossy conducting medium, the wave front normalis parallel to the tangent of the conducting image ground plane 130 atz=−d/2, and not at the boundary interface between Regions 1 and 2.

Consider the case illustrated in FIG. 8A where the lossy conductingmedium 203 is a finitely conducting Earth 133 with a physical boundary136. The finitely conducting Earth 133 may be replaced by a perfectlyconducting image ground plane 139 as shown in FIG. 8B, which is locatedat a complex depth z₁ below the physical boundary 136. This equivalentrepresentation exhibits the same impedance when looking down into theinterface at the physical boundary 136. The equivalent representation ofFIG. 8B can be modeled as an equivalent transmission line, as shown inFIG. 8C. The cross-section of the equivalent structure is represented asa (z-directed) end-loaded transmission line, with the impedance of theperfectly conducting image plane being a short circuit (z_(s)=0). Thedepth z₁ can be determined by equating the TEM wave impedance lookingdown at the Earth to an image ground plane impedance z_(in) seen lookinginto the transmission line of FIG. 8C.

In the case of FIG. 8A, the propagation constant and wave intrinsicimpedance in the upper region (air) 142 are

$\begin{matrix}{{\gamma_{o} = {{j\; \omega \sqrt{\mu_{o}ɛ_{o}}} = {0 + {j\; \beta_{o}}}}},{and}} & (55) \\{z_{o} = {\frac{j\; {\omega\mu}_{o}}{\gamma_{o}} = {\sqrt{\frac{\mu_{o}}{ɛ_{o}}}.}}} & (56)\end{matrix}$

In the lossy Earth 133, the propagation constant and wave intrinsicimpedance are

$\begin{matrix}{{\gamma_{e} = \sqrt{j\; \omega \; {\mu_{1}\left( {\sigma_{1} + {j\; \omega \; ɛ_{1}}} \right)}}},{and}} & (57) \\{Z_{e} = {\frac{j\; \omega \; \mu_{1}}{\gamma_{e}}.}} & (58)\end{matrix}$

For normal incidence, the equivalent representation of FIG. 8B isequivalent to a TEM transmission line whose characteristic impedance isthat of air (z₀), with propagation constant of γ₀, and whose length isz₁. As such, the image ground plane impedance Z_(in) seen at theinterface for the shorted transmission line of FIG. 8C is given by

Z _(in) =Z ₀ tan h(γ₀ z ₁).  (59)

Equating the image ground plane impedance Z_(in) associated with theequivalent model of FIG. 8C to the normal incidence wave impedance ofFIG. 8A and solving for z₁ gives the distance to a short circuit (theperfectly conducting image ground plane 139) as

$\begin{matrix}{{z_{1} = {{\frac{1}{\gamma_{o}}{\tanh^{- 1}\left( \frac{Z_{e}}{Z_{o}} \right)}} = {{\frac{1}{\gamma_{o}}{\tanh^{- 1}\left( \frac{\gamma_{o}}{\gamma_{e}} \right)}} \approx \frac{1}{\gamma_{e}}}}},} & (60)\end{matrix}$

where only the first term of the series expansion for the inversehyperbolic tangent is considered for this approximation. Note that inthe air region 142, the propagation constant is γ₀=jβ₀, so Z_(in)=jZ₀tan β₀z₁ (which is a purely imaginary quantity for a real z₁), but z_(e)is a complex value if σ≠0. Therefore, Z_(in)=Z_(e) only when z₁ is acomplex distance.

Since the equivalent representation of FIG. 8B includes a perfectlyconducting image ground plane 139, the image depth for a charge orcurrent lying at the surface of the Earth (physical boundary 136) isequal to distance z₁ on the other side of the image ground plane 139, ord=2×z₁ beneath the Earth's surface (which is located at z=0). Thus, thedistance to the perfectly conducting image ground plane 139 can beapproximated by

$\begin{matrix}{d = {{2z_{1}} \approx {\frac{2}{\gamma_{e}}.}}} & (61)\end{matrix}$

Additionally, the “image charge” will be “equal and opposite” to thereal charge, so the potential of the perfectly conducting image groundplane 139 at depth z₁=−d/2 will be zero.

If a charge Q₁ is elevated a distance H₁ above the surface of the Earthas illustrated in FIG. 3, then the image charge Q₁′ resides at a complexdistance of D₁=d+H₁ below the surface, or a complex distance of d/2+H₁below the image ground plane 130. The guided surface waveguide probe 200b of FIG. 7 can be modeled as an equivalent single-wire transmissionline image plane model that can be based upon the perfectly conductingimage ground plane 139 of FIG. 8B. FIG. 9A shows an example of theequivalent single-wire transmission line image plane model, and FIG. 9Billustrates an example of the equivalent classic transmission linemodel, including the shorted transmission line of FIG. 8C.

In the equivalent image plane models of FIGS. 9A and 9B, Φ=θ_(y)+θ_(c)is the traveling wave phase delay of the guided surface waveguide probe200 referenced to Earth 133 (or the lossy conducting medium 203),θ_(c)=β_(p)H is the electrical length of the coil 215 (FIG. 7), ofphysical length H, expressed in degrees, θ_(y)=β_(w)h_(w) is theelectrical length of the vertical feed line conductor 221 (FIG. 7), ofphysical length h_(w), expressed in degrees, and θ_(d)=β₀d/2 is thephase shift between the image ground plane 139 and the physical boundary136 of the Earth 133 (or lossy conducting medium 203). In the example ofFIGS. 9A and 9B, Z_(w) is the characteristic impedance of the elevatedvertical feed line conductor 221 in ohms, Z_(c) is the characteristicimpedance of the coil 215 in ohms, and Z₀ is the characteristicimpedance of free space.

At the base of the guided surface waveguide probe 200, the impedanceseen “looking up” into the structure is Z_(↑)=Z_(base). With a loadimpedance of:

$\begin{matrix}{{Z_{L} = \frac{1}{j\; \omega \; C_{T}}},} & (62)\end{matrix}$

where C_(T) is the self-capacitance of the charge terminal T₁, theimpedance seen “looking up” into the vertical feed line conductor 221(FIG. 7) is given by:

$\begin{matrix}{{Z_{2} = {{Z_{W}\frac{Z_{L} + {Z_{w}\mspace{14mu} {\tanh \left( {j\; \beta_{w}h_{w}} \right)}}}{Z_{w} + {Z_{L}\mspace{14mu} {\tanh \left( {j\; \beta_{w}h_{w}} \right)}}}} = {Z_{W}\frac{Z_{L} + {Z_{w}\mspace{14mu} {\tanh \left( {j\; \theta_{y}} \right)}}}{Z_{w} + {Z_{L}\mspace{14mu} {\tanh \left( {j\; \theta_{y}} \right)}}}}}},} & (63)\end{matrix}$

and the impedance seen “looking up” into the coil 215 (FIG. 7) is givenby:

$\begin{matrix}{Z_{base} = {{Z_{c}\frac{Z_{2} + {Z_{c}\mspace{14mu} {\tanh \left( {j\; \beta_{p}H} \right)}}}{Z_{c} + {Z_{2}\mspace{14mu} {\tanh \left( {j\; \beta_{p}H} \right)}}}} = {Z_{c}{\frac{Z_{2} + {Z_{c}\mspace{14mu} {\tanh \left( {j\; \theta_{c}} \right)}}}{Z_{c} + {Z_{2}\mspace{14mu} {\tanh \left( {j\; \theta_{c}} \right)}}}.}}}} & (64)\end{matrix}$

At the base of the guided surface waveguide probe 200, the impedanceseen “looking down” into the lossy conducting medium 203 isZ_(↓)=Z_(in), which is given by:

$\begin{matrix}{{Z_{in} = {{Z_{o}\frac{Z_{s} + {Z_{o}\mspace{14mu} {\tanh \left\lbrack {j\; {\beta_{o}\left( {d\text{/}2} \right)}} \right\rbrack}}}{Z_{o} + {Z_{s}\mspace{14mu} {\tanh \left\lbrack {j\; {\beta_{o}\left( {d\text{/}2} \right)}} \right\rbrack}}}} = {Z_{o}\mspace{14mu} {\tanh \left( {j\; \theta_{d}} \right)}}}},} & (65)\end{matrix}$

where Z_(s)=0.

Neglecting losses, the equivalent image plane model can be tuned toresonance when Z_(↓)+Z_(↑)=0 at the physical boundary 136. Or, in thelow loss case, X_(↓)+X_(↑)=0 at the physical boundary 136, where X isthe corresponding reactive component. Thus, the impedance at thephysical boundary 136 “looking up” into the guided surface waveguideprobe 200 is the conjugate of the impedance at the physical boundary 136“looking down” into the lossy conducting medium 203. By adjusting theload impedance Z_(L) of the charge terminal T₁ while maintaining thetraveling wave phase delay Φ equal to the angle of the media's wave tiltΨ, so that Φ=Ψ, which improves and/or maximizes coupling of the probe'selectric field to a guided surface waveguide mode along the surface ofthe lossy conducting medium 203 (e.g., Earth), the equivalent imageplane models of FIGS. 9A and 9B can be tuned to resonance with respectto the image ground plane 139. In this way, the impedance of theequivalent complex image plane model is purely resistive, whichmaintains a superposed standing wave on the probe structure thatmaximizes the voltage and elevated charge on terminal T₁, and byequations (1)-(3) and (16) maximizes the propagating surface wave.

It follows from the Hankel solutions, that the guided surface waveexcited by the guided surface waveguide probe 200 is an outwardpropagating traveling wave. The source distribution along the feednetwork 209 between the charge terminal T₁ and the ground stake 218 ofthe guided surface waveguide probe 200 (FIGS. 3 and 7) is actuallycomposed of a superposition of a traveling wave plus a standing wave onthe structure. With the charge terminal T₁ positioned at or above thephysical height h_(p), the phase delay of the traveling wave movingthrough the feed network 209 is matched to the angle of the wave tiltassociated with the lossy conducting medium 203. This mode-matchingallows the traveling wave to be launched along the lossy conductingmedium 203. Once the phase delay has been established for the travelingwave, the load impedance Z_(L) of the charge terminal T₁ is adjusted tobring the probe structure into standing wave resonance with respect tothe image ground plane (130 of FIG. 3 or 139 of FIG. 8), which is at acomplex depth of −d/2. In that case, the impedance seen from the imageground plane has zero reactance and the charge on the charge terminal T₁is maximized.

The distinction between the traveling wave phenomenon and standing wavephenomena is that (1) the phase delay of traveling waves (θ=βd) on asection of transmission line of length d (sometimes called a “delayline”) is due to propagation time delays; whereas (2) theposition-dependent phase of standing waves (which are composed offorward and backward propagating waves) depends on both the line lengthpropagation time delay and impedance transitions at interfaces betweenline sections of different characteristic impedances. In addition to thephase delay that arises due to the physical length of a section oftransmission line operating in sinusoidal steady-state, there is anextra reflection coefficient phase at impedance discontinuities that isdue to the ratio of Z_(oa)/Z_(ob), where Z_(oa) and Z_(ob) are thecharacteristic impedances of two sections of a transmission line suchas, e.g., a helical coil section of characteristic impedanceZ_(oa)=Z_(c) (FIG. 9B) and a straight section of vertical feed lineconductor of characteristic impedance Z_(ob)=Z_(w) (FIG. 9B).

As a result of this phenomenon, two relatively short transmission linesections of widely differing characteristic impedance may be used toprovide a very large phase shift. For example, a probe structurecomposed of two sections of transmission line, one of low impedance andone of high impedance, together totaling a physical length of, say,0.05λ, may be fabricated to provide a phase shift of 90° which isequivalent to a 0.25λ resonance. This is due to the large jump incharacteristic impedances. In this way, a physically short probestructure can be electrically longer than the two physical lengthscombined. This is illustrated in FIGS. 9A and 9B, where thediscontinuities in the impedance ratios provide large jumps in phase.The impedance discontinuity provides a substantial phase shift where thesections are joined together.

Referring to FIG. 10, shown is a flow chart 150 illustrating an exampleof adjusting a guided surface waveguide probe 200 (FIGS. 3 and 7) tosubstantially mode-match to a guided surface waveguide mode on thesurface of the lossy conducting medium, which launches a guided surfacetraveling wave along the surface of a lossy conducting medium 203 (FIG.3). Beginning with 153, the charge terminal T₁ of the guided surfacewaveguide probe 200 is positioned at a defined height above a lossyconducting medium 203. Utilizing the characteristics of the lossyconducting medium 203 and the operating frequency of the guided surfacewaveguide probe 200, the Hankel crossover distance can also be found byequating the magnitudes of Equations (20b) and (21) for −jγρ, andsolving for R_(x) as illustrated by FIG. 4. The complex index ofrefraction (n) can be determined using Equation (41), and the complexBrewster angle (θ_(i,B)) can then be determined from Equation (42). Thephysical height (h_(p)) of the charge terminal T₁ can then be determinedfrom Equation (44). The charge terminal T₁ should be at or higher thanthe physical height (h_(p)) in order to excite the far-out component ofthe Hankel function. This height relationship is initially consideredwhen launching surface waves. To reduce or minimize the bound charge onthe charge terminal T₁, the height should be at least four times thespherical diameter (or equivalent spherical diameter) of the chargeterminal T₁.

At 156, the electrical phase delay Φ of the elevated charge Q₁ on thecharge terminal T₁ is matched to the complex wave tilt angle Ψ. Thephase delay (θ_(c)) of the helical coil and/or the phase delay (θ_(y))of the vertical feed line conductor can be adjusted to make Φ equal tothe angle (Ψ) of the wave tilt (W). Based on Equation (31), the angle(Ψ) of the wave tilt can be determined from:

$\begin{matrix}{W = {\frac{E_{\rho}}{E_{z}} = {\frac{1}{\tan \mspace{14mu} \theta_{i,B}} = {\frac{1}{n} = \left| W \middle| {e^{j\; \Psi}.} \right.}}}} & (66)\end{matrix}$

The electrical phase Φ can then be matched to the angle of the wavetilt. This angular (or phase) relationship is next considered whenlaunching surface waves. For example, the electrical phase delayΦ=θ_(c)+θ_(y) can be adjusted by varying the geometrical parameters ofthe coil 215 (FIG. 7) and/or the length (or height) of the vertical feedline conductor 221 (FIG. 7). By matching Φ=Ψ, an electric field can beestablished at or beyond the Hankel crossover distance (R_(x)) with acomplex Brewster angle at the boundary interface to excite the surfacewaveguide mode and launch a traveling wave along the lossy conductingmedium 203.

Next at 159, the load impedance of the charge terminal T₁ is tuned toresonate the equivalent image plane model of the guided surfacewaveguide probe 200. The depth (d/2) of the conducting image groundplane 139 of FIGS. 9A and 9B (or 130 of FIG. 3) can be determined usingEquations (52), (53) and (54) and the values of the lossy conductingmedium 203 (e.g., the Earth), which can be measured. Using that depth,the phase shift (θ_(d)) between the image ground plane 139 and thephysical boundary 136 of the lossy conducting medium 203 can bedetermined using θ_(d)=β₀ d/2. The impedance (Z_(in)) as seen “lookingdown” into the lossy conducting medium 203 can then be determined usingEquation (65). This resonance relationship can be considered to maximizethe launched surface waves.

Based upon the adjusted parameters of the coil 215 and the length of thevertical feed line conductor 221, the velocity factor, phase delay, andimpedance of the coil 215 and vertical feed line conductor 221 can bedetermined using Equations (45) through (51). In addition, theself-capacitance (C_(T)) of the charge terminal T₁ can be determinedusing, e.g., Equation (24). The propagation factor (β_(p)) of the coil215 can be determined using Equation (35) and the propagation phaseconstant (β_(w)) for the vertical feed line conductor 221 can bedetermined using Equation (49). Using the self-capacitance and thedetermined values of the coil 215 and vertical feed line conductor 221,the impedance (Z_(base)) of the guided surface waveguide probe 200 asseen “looking up” into the coil 215 can be determined using Equations(62), (63) and (64).

The equivalent image plane model of the guided surface waveguide probe200 can be tuned to resonance by adjusting the load impedance Z_(L) suchthat the reactance component X_(base) of Z_(base) cancels out thereactance component X_(in) of Z_(in), or X_(base) X_(in)=0. Thus, theimpedance at the physical boundary 136 “looking up” into the guidedsurface waveguide probe 200 is the conjugate of the impedance at thephysical boundary 136 “looking down” into the lossy conducting medium203. The load impedance Z_(L) can be adjusted by varying the capacitance(C_(T)) of the charge terminal T₁ without changing the electrical phasedelay Φ=θ_(c)+θ_(y) of the charge terminal T₁. An iterative approach maybe taken to tune the load impedance Z_(L) for resonance of theequivalent image plane model with respect to the conducting image groundplane 139 (or 130). In this way, the coupling of the electric field to aguided surface waveguide mode along the surface of the lossy conductingmedium 203 (e.g., Earth) can be improved and/or maximized.

This may be better understood by illustrating the situation with anumerical example. Consider a guided surface waveguide probe 200comprising a top-loaded vertical stub of physical height h_(p) with acharge terminal T₁ at the top, where the charge terminal T₁ is excitedthrough a helical coil and vertical feed line conductor at anoperational frequency (f₀) of 1.85 MHz. With a height (H₁) of 16 feetand the lossy conducting medium 203 (e.g., Earth) having a relativepermittivity of ε_(r)=15 and a conductivity of σ₁=0.010 mhos/m, severalsurface wave propagation parameters can be calculated for f₀=1.850 MHz.Under these conditions, the Hankel crossover distance can be found to beR_(x)=54.5 feet with a physical height of h_(p)=5.5 feet, which is wellbelow the actual height of the charge terminal T₁. While a chargeterminal height of H₁=5.5 feet could have been used, the taller probestructure reduced the bound capacitance, permitting a greater percentageof free charge on the charge terminal T₁ providing greater fieldstrength and excitation of the traveling wave.

The wave length can be determined as:

$\begin{matrix}{{\lambda_{o} = {\frac{c}{f_{o}} = {162.162\mspace{14mu} {meters}}}},} & (67)\end{matrix}$

where c is the speed of light. The complex index of refraction is:

n=√{square root over (ε_(r) −jx)}=7.529−j6.546,  (68)

from Equation (41), where x=σ₁/ωε₀ with ω=2πf₀, and the complex Brewsterangle is:

θ_(i,B)=arctan(√{square root over (ε_(r) −jx)})=85.6−j3.744°.  (69)

from Equation (42). Using Equation (66), the wave tilt values can bedetermined to be:

$\begin{matrix}{W = {\frac{1}{\tan \mspace{14mu} \theta_{i,B}} = {\frac{1}{n} = {\left| W \middle| e^{j\; \Psi} \right. = {0.101{e^{j\; 40.614{^\circ}}.}}}}}} & (70)\end{matrix}$

Thus, the helical coil can be adjusted to match Φ=Ψ=40.614°

The velocity factor of the vertical feed line conductor (approximated asa uniform cylindrical conductor with a diameter of 0.27 inches) can begiven as V_(w)≈0.93. Since h_(p)«λ₀, the propagation phase constant forthe vertical feed line conductor can be approximated as:

$\begin{matrix}{\beta_{w} = {\frac{2\pi}{\lambda_{w}} = {\frac{2\pi}{V_{w}\lambda_{0}} = {0.042\mspace{14mu} {m^{- 1}.}}}}} & (71)\end{matrix}$

From Equation (49) the phase delay of the vertical feed line conductoris:

θ_(y)=β_(w) h _(w)≈β_(w) h _(p)=11.640°.  (72)

By adjusting the phase delay of the helical coil so thatθ_(c)=28.974°=40.614°−11.640°, Φ will equal Ψ to match the guidedsurface waveguide mode. To illustrate the relationship between Φ and Ψ,FIG. 11 shows a plot of both over a range of frequencies. As both Φ andΨ are frequency dependent, it can be seen that their respective curvescross over each other at approximately 1.85 MHz.

For a helical coil having a conductor diameter of 0.0881 inches, a coildiameter (D) of 30 inches and a turn-to-turn spacing (s) of 4 inches,the velocity factor for the coil can be determined using Equation (45)as:

$\begin{matrix}{{V_{f} = {\frac{1}{\sqrt{1 + {20\left( \frac{D}{s} \right)^{2.5}\left( \frac{D}{\lambda_{o}} \right)^{0.5}}}} = 0.069}},} & (73)\end{matrix}$

and the propagation factor from Equation (35) is:

$\begin{matrix}{\beta_{p} = {\frac{2\pi}{V_{f}\lambda_{0}} = {0.564\mspace{14mu} {m^{- 1}.}}}} & (74)\end{matrix}$

With θ_(c)=28.974°, the axial length of the solenoidal helix (H) can bedetermined using Equation (46) such that:

$\begin{matrix}{H = {\frac{\theta_{c}}{\beta_{p}} = {35.2732\mspace{14mu} {{inches}.}}}} & (75)\end{matrix}$

This height determines the location on the helical coil where thevertical feed line conductor is connected, resulting in a coil with8.818 turns (N=H/s).

With the traveling wave phase delay of the coil and vertical feed lineconductor adjusted to match the wave tilt angle (Φ=θ_(c)+θ_(y)=Ψ), theload impedance (Z_(L)) of the charge terminal T₁ can be adjusted forstanding wave resonance of the equivalent image plane model of theguided surface wave probe 200. From the measured permittivity,conductivity and permeability of the Earth, the radial propagationconstant can be determined using Equation (57)

γe=√{square root over (jωu ₁(σ₁ +jωε ₁))}=0.25+j0.292 m⁻¹,  (76)

And the complex depth of the conducting image ground plane can beapproximated from Equation (52) as:

$\begin{matrix}{{{d \approx \frac{2}{\gamma_{e}}} = {3.364 + {j\mspace{14mu} 3.963\mspace{14mu} {meters}}}},} & (77)\end{matrix}$

with a corresponding phase shift between the conducting image groundplane and the physical boundary of the Earth given by:

θ_(d)=β₀(d/2)=4.015−j4.73°.  (78)

Using Equation (65), the impedance seen “looking down” into the lossyconducting medium 203 (i.e., Earth) can be determined as:

Z _(in) =Z ₀ tan h(jθ _(d))=R _(in) +jX _(in)=31.191+j26.27 ohms.  (79)

By matching the reactive component (X_(in)) seen “looking down” into thelossy conducting medium 203 with the reactive component (X_(base)) seen“looking up” into the guided surface wave probe 200, the coupling intothe guided surface waveguide mode may be maximized. This can beaccomplished by adjusting the capacitance of the charge terminal T₁without changing the traveling wave phase delays of the coil andvertical feed line conductor. For example, by adjusting the chargeterminal capacitance (C_(T)) to 61.8126 pF, the load impedance fromEquation (62) is:

$\begin{matrix}{{Z_{L} = {\frac{1}{j\; \omega \; C_{T}} = {{- j}\; 1392\mspace{14mu} {ohms}}}},} & (80)\end{matrix}$

and the reactive components at the boundary are matched.

Using Equation (51), the impedance of the vertical feed line conductor(having a diameter (2a) of 0.27 inches) is given as

$\begin{matrix}{{Z_{w} = {{138\mspace{14mu} {\log \left( \frac{1.123\mspace{14mu} V_{w}\lambda_{0}}{2\pi \; a} \right)}} = {537.534\mspace{14mu} {ohms}}}},} & (81)\end{matrix}$

and the impedance seen “looking up” into the vertical feed lineconductor is given by Equation (63) as:

$\begin{matrix}{{Z_{2} = {{Z_{W}\frac{Z_{L} + {Z_{w}\mspace{14mu} {\tanh \left( {j\; \theta_{y}} \right)}}}{Z_{w} + {Z_{L}\mspace{14mu} {\tanh \left( {j\; \theta_{y}} \right)}}}} = {{- j}\mspace{14mu} 835.438\mspace{14mu} {{ohms}.}}}}\;} & (82)\end{matrix}$

Using Equation (47), the characteristic impedance of the helical coil isgiven as

$\begin{matrix}{{Z_{c} = {{\frac{60}{V_{f}}\left\lbrack {{\ln \left( \frac{V_{f}\lambda_{0}}{D} \right)} - 1.027} \right\rbrack} = {1446\mspace{14mu} {ohms}}}},} & (83)\end{matrix}$

and the impedance seen “looking up” into the coil at the base is givenby Equation (64) as:

$\begin{matrix}{Z_{base} = {{Z_{c}\frac{Z_{2} + {Z_{c}\mspace{14mu} {\tanh \left( {j\; \theta_{c}} \right)}}}{Z_{c} + {Z_{2}\mspace{14mu} {\tanh \left( {j\; \theta_{c}} \right)}}}} = {{- j}\mspace{14mu} 26.271\mspace{14mu} {{ohms}.}}}} & (84)\end{matrix}$

When compared to the solution of Equation (79), it can be seen that thereactive components are opposite and approximately equal, and thus areconjugates of each other. Thus, the impedance (Z_(ip)) seen “looking up”into the equivalent image plane model of FIGS. 9A and 9B from theperfectly conducting image ground plane is only resistive orZ_(ip)=R+j0.

When the electric fields produced by a guided surface waveguide probe200 (FIG. 3) are established by matching the traveling wave phase delayof the feed network to the wave tilt angle and the probe structure isresonated with respect to the perfectly conducting image ground plane atcomplex depth z=−d/2, the fields are substantially mode-matched to aguided surface waveguide mode on the surface of the lossy conductingmedium, a guided surface traveling wave is launched along the surface ofthe lossy conducting medium. As illustrated in FIG. 1, the guided fieldstrength curve 103 of the guided electromagnetic field has acharacteristic exponential decay of e^(−αd)/√{square root over (d)} andexhibits a distinctive knee 109 on the log-log scale.

In summary, both analytically and experimentally, the traveling wavecomponent on the structure of the guided surface waveguide probe 200 hasa phase delay (Φ) at its upper terminal that matches the angle (Ψ) ofthe wave tilt of the surface traveling wave (Φ=Ψ). Under this condition,the surface waveguide may be considered to be “mode-matched”.Furthermore, the resonant standing wave component on the structure ofthe guided surface waveguide probe 200 has a V_(MAX) at the chargeterminal T₁ and a V_(MIN) down at the image plane 139 (FIG. 8B) whereZ_(ip)=R_(ip)+j 0 at a complex depth of z=−d/2, not at the connection atthe physical boundary 136 of the lossy conducting medium 203 (FIG. 8B).Lastly, the charge terminal T₁ is of sufficient height H₁ of FIG. 3(h≥R_(x) tan ψ_(i,B)) so that electromagnetic waves incident onto thelossy conducting medium 203 at the complex Brewster angle do so out at adistance (≥R_(x)) where the 1/√{square root over (r)} term ispredominant. Receive circuits can be utilized with one or more guidedsurface waveguide probes to facilitate wireless transmission and/orpower delivery systems.

Referring back to FIG. 3, operation of a guided surface waveguide probe200 may be controlled to adjust for variations in operational conditionsassociated with the guided surface waveguide probe 200. For example, anadaptive probe control system 230 can be used to control the feednetwork 209 and/or the charge terminal T₁ to control the operation ofthe guided surface waveguide probe 200. Operational conditions caninclude, but are not limited to, variations in the characteristics ofthe lossy conducting medium 203 (e.g., conductivity a and relativepermittivity ε_(r)), variations in field strength and/or variations inloading of the guided surface waveguide probe 200. As can be seen fromEquations (31), (41) and (42), the index of refraction (n), the complexBrewster angle (θ_(i,B)), and the wave tilt (|W|e^(jΨ)) can be affectedby changes in soil conductivity and permittivity resulting from, e.g.,weather conditions.

Equipment such as, e.g., conductivity measurement probes, permittivitysensors, ground parameter meters, field meters, current monitors and/orload receivers can be used to monitor for changes in the operationalconditions and provide information about current operational conditionsto the adaptive probe control system 230. The probe control system 230can then make one or more adjustments to the guided surface waveguideprobe 200 to maintain specified operational conditions for the guidedsurface waveguide probe 200. For instance, as the moisture andtemperature vary, the conductivity of the soil will also vary.Conductivity measurement probes and/or permittivity sensors may belocated at multiple locations around the guided surface waveguide probe200. Generally, it would be desirable to monitor the conductivity and/orpermittivity at or about the Hankel crossover distance R_(x) for theoperational frequency. Conductivity measurement probes and/orpermittivity sensors may be located at multiple locations (e.g., in eachquadrant) around the guided surface waveguide probe 200.

The conductivity measurement probes and/or permittivity sensors can beconfigured to evaluate the conductivity and/or permittivity on aperiodic basis and communicate the information to the probe controlsystem 230. The information may be communicated to the probe controlsystem 230 through a network such as, but not limited to, a LAN, WLAN,cellular network, or other appropriate wired or wireless communicationnetwork. Based upon the monitored conductivity and/or permittivity, theprobe control system 230 may evaluate the variation in the index ofrefraction (n), the complex Brewster angle (θ_(i,B)), and/or the wavetilt (|W|e^(jΨ)) and adjust the guided surface waveguide probe 200 tomaintain the phase delay (Φ) of the feed network 209 equal to the wavetilt angle (Ψ) and/or maintain resonance of the equivalent image planemodel of the guided surface waveguide probe 200. This can beaccomplished by adjusting, e.g., θ_(y), θ_(c) and/or C_(T). Forinstance, the probe control system 230 can adjust the self-capacitanceof the charge terminal T₁ and/or the phase delay (θ_(y), θ_(c)) appliedto the charge terminal T₁ to maintain the electrical launchingefficiency of the guided surface wave at or near its maximum. Forexample, the self-capacitance of the charge terminal T₁ can be varied bychanging the size of the terminal. The charge distribution can also beimproved by increasing the size of the charge terminal T₁, which canreduce the chance of an electrical discharge from the charge terminalT₁. In other embodiments, the charge terminal T₁ can include a variableinductance that can be adjusted to change the load impedance Z_(L). Thephase applied to the charge terminal T₁ can be adjusted by varying thetap position on the coil 215 (FIG. 7), and/or by including a pluralityof predefined taps along the coil 215 and switching between thedifferent predefined tap locations to maximize the launching efficiency.

Field or field strength (FS) meters may also be distributed about theguided surface waveguide probe 200 to measure field strength of fieldsassociated with the guided surface wave. The field or FS meters can beconfigured to detect the field strength and/or changes in the fieldstrength (e.g., electric field strength) and communicate thatinformation to the probe control system 230. The information may becommunicated to the probe control system 230 through a network such as,but not limited to, a LAN, WLAN, cellular network, or other appropriatecommunication network. As the load and/or environmental conditionschange or vary during operation, the guided surface waveguide probe 200may be adjusted to maintain specified field strength(s) at the FS meterlocations to ensure appropriate power transmission to the receivers andthe loads they supply.

For example, the phase delay (Φ=θ_(y)+θ_(c)) applied to the chargeterminal T₁ can be adjusted to match the wave tilt angle (Ψ). Byadjusting one or both phase delays, the guided surface waveguide probe200 can be adjusted to ensure the wave tilt corresponds to the complexBrewster angle. This can be accomplished by adjusting a tap position onthe coil 215 (FIG. 7) to change the phase delay supplied to the chargeterminal T₁. The voltage level supplied to the charge terminal T₁ canalso be increased or decreased to adjust the electric field strength.This may be accomplished by adjusting the output voltage of theexcitation source 212 or by adjusting or reconfiguring the feed network209. For instance, the position of the tap 227 (FIG. 7) for the ACsource 212 can be adjusted to increase the voltage seen by the chargeterminal Maintaining field strength levels within predefined ranges canimprove coupling by the receivers, reduce ground current losses, andavoid interference with transmissions from other guided surfacewaveguide probes 200.

The probe control system 230 can be implemented with hardware, firmware,software executed by hardware, or a combination thereof. For example,the probe control system 230 can include processing circuitry includinga processor and a memory, both of which can be coupled to a localinterface such as, for example, a data bus with an accompanyingcontrol/address bus as can be appreciated by those with ordinary skillin the art. A probe control application may be executed by the processorto adjust the operation of the guided surface waveguide probe 200 basedupon monitored conditions. The probe control system 230 can also includeone or more network interfaces for communicating with the variousmonitoring devices. Communications can be through a network such as, butnot limited to, a LAN, WLAN, cellular network, or other appropriatecommunication network. The probe control system 230 may comprise, forexample, a computer system such as a server, desktop computer, laptop,or other system with like capability.

Referring back to the example of FIG. 5A, the complex angle trigonometryis shown for the ray optic interpretation of the incident electric field(E) of the charge terminal T₁ with a complex Brewster angle (θ_(i,B)) atthe Hankel crossover distance (R_(x)). Recall that, for a lossyconducting medium, the Brewster angle is complex and specified byequation (38). Electrically, the geometric parameters are related by theelectrical effective height (h_(eff)) of the charge terminal T₁ byequation (39). Since both the physical height (h_(p)) and the Hankelcrossover distance (R_(x)) are real quantities, the angle of the desiredguided surface wave tilt at the Hankel crossover distance (W_(Rx)) isequal to the phase (Φ) of the complex effective height (h_(eff)). Withthe charge terminal T₁ positioned at the physical height h_(p) andexcited with a charge having the appropriate phase Φ, the resultingelectric field is incident with the lossy conducting medium boundaryinterface at the Hankel crossover distance R_(x), and at the Brewsterangle. Under these conditions, the guided surface waveguide mode can beexcited without reflection or substantially negligible reflection.

However, Equation (39) means that the physical height of the guidedsurface waveguide probe 200 can be relatively small. While this willexcite the guided surface waveguide mode, this can result in an undulylarge bound charge with little free charge. To compensate, the chargeterminal T₁ can be raised to an appropriate elevation to increase theamount of free charge. As one example rule of thumb, the charge terminalT₁ can be positioned at an elevation of about 4-5 times (or more) theeffective diameter of the charge terminal T₁. FIG. 6 illustrates theeffect of raising the charge terminal T₁ above the physical height(h_(p)) shown in FIG. 5A. The increased elevation causes the distance atwhich the wave tilt is incident with the lossy conductive medium to movebeyond the Hankel crossover point 121 (FIG. 5A). To improve coupling inthe guided surface waveguide mode, and thus provide for a greaterlaunching efficiency of the guided surface wave, a lower compensationterminal T₂ can be used to adjust the total effective height (h_(TE)) ofthe charge terminal T₁ such that the wave tilt at the Hankel crossoverdistance is at the Brewster angle.

Referring to FIG. 12, shown is an example of a guided surface waveguideprobe 200 c that includes an elevated charge terminal T₁ and a lowercompensation terminal T₂ that are arranged along a vertical axis z thatis normal to a plane presented by the lossy conducting medium 203. Inthis respect, the charge terminal T₁ is placed directly above thecompensation terminal T₂ although it is possible that some otherarrangement of two or more charge and/or compensation terminals TN canbe used. The guided surface waveguide probe 200 c is disposed above alossy conducting medium 203 according to an embodiment of the presentdisclosure. The lossy conducting medium 203 makes up Region 1 with asecond medium 206 that makes up Region 2 sharing a boundary interfacewith the lossy conducting medium 203.

The guided surface waveguide probe 200 c includes a feed network 209that couples an excitation source 212 to the charge terminal T₁ and thecompensation terminal T₂. According to various embodiments, charges Q₁and Q₂ can be imposed on the respective charge and compensationterminals T₁ and T₂, depending on the voltages applied to terminals T₁and T₂ at any given instant. I₁ is the conduction current feeding thecharge Q₁ on the charge terminal T₁ via the terminal lead, and I₂ is theconduction current feeding the charge Q₂ on the compensation terminal T₂via the terminal lead.

According to the embodiment of FIG. 12, the charge terminal T₁ ispositioned over the lossy conducting medium 203 at a physical height H₁,and the compensation terminal T₂ is positioned directly below T₁ alongthe vertical axis z at a physical height H₂, where H₂ is less than H₁.The height h of the transmission structure may be calculated as h=H₁−H₂.The charge terminal T₁ has an isolated (or self) capacitance C₁, and thecompensation terminal T₂ has an isolated (or self) capacitance C₂. Amutual capacitance C_(M) can also exist between the terminals T₁ and T₂depending on the distance therebetween. During operation, charges Q₁ andQ₂ are imposed on the charge terminal T₁ and the compensation terminalT₂, respectively, depending on the voltages applied to the chargeterminal T₁ and the compensation terminal T₂ at any given instant.

Referring next to FIG. 13, shown is a ray optics interpretation of theeffects produced by the elevated charge Q₁ on charge terminal T₁ andcompensation terminal T₂ of FIG. 12. With the charge terminal T₁elevated to a height where the ray intersects with the lossy conductivemedium at the Brewster angle at a distance greater than the Hankelcrossover point 121 as illustrated by line 163, the compensationterminal T₂ can be used to adjust h_(TE) by compensating for theincreased height. The effect of the compensation terminal T₂ is toreduce the electrical effective height of the guided surface waveguideprobe (or effectively raise the lossy medium interface) such that thewave tilt at the Hankel crossover distance is at the Brewster angle asillustrated by line 166.

The total effective height can be written as the superposition of anupper effective height (h_(UE)) associated with the charge terminal T₁and a lower effective height (h_(LE)) associated with the compensationterminal T₂ such that

h _(TE) =h _(UE) +h _(LE) =h _(p) e ^(j(βh) ^(p) ^(+Φ) ^(U) ⁾ +h _(d) e^(j(βh) ^(d) ^(+Φ) ^(L) ⁾ =R _(x) ×W,  (85)

where Φ_(U) is the phase delay applied to the upper charge terminal T₁,Φ_(L) is the phase delay applied to the lower compensation terminal T₂,β=2π/λ_(p) is the propagation factor from Equation (35), h_(p) is thephysical height of the charge terminal T₁ and h_(d) is the physicalheight of the compensation terminal T₂. If extra lead lengths are takeninto consideration, they can be accounted for by adding the chargeterminal lead length z to the physical height h_(p) of the chargeterminal T₁ and the compensation terminal lead length y to the physicalheight h_(d) of the compensation terminal T₂ as shown in

h _(TE)=(h _(p) +z)e ^(j(βh) ^(p) ^(+z)+Φ) ^(U) ⁾+(h _(d) +y)e ^(j(βh)^(d) ^(+y)Φ) ^(L) ⁾ =R _(x) ×W.  (86)

The lower effective height can be used to adjust the total effectiveheight (h_(TE)) to equal the complex effective height (h_(eff)) of FIG.5A.

Equations (85) or (86) can be used to determine the physical height ofthe lower disk of the compensation terminal T₂ and the phase angles tofeed the terminals in order to obtain the desired wave tilt at theHankel crossover distance. For example, Equation (86) can be rewrittenas the phase shift applied to the charge terminal T₁ as a function ofthe compensation terminal height (h_(d)) to give

$\begin{matrix}{{\Phi_{U}\left( h_{d} \right)} = {{- {\beta \left( {h_{p} + z} \right)}} - {j\mspace{14mu} {{\ln \left( \frac{{R_{x} \times W} - {\left( {h_{d} + y} \right)e^{j{({{\beta \; h_{d}} + {\beta \; y} + \Phi_{L}})}}}}{\left( {h_{p} + z} \right)} \right)}.}}}} & (87)\end{matrix}$

To determine the positioning of the compensation terminal T₂, therelationships discussed above can be utilized. First, the totaleffective height (h_(TE)) is the superposition of the complex effectiveheight (h_(UE)) of the upper charge terminal T₁ and the complexeffective height (h_(LE)) of the lower compensation terminal T₂ asexpressed in Equation (86). Next, the tangent of the angle of incidencecan be expressed geometrically as

$\begin{matrix}{{{\tan \mspace{14mu} \psi_{E}} = \frac{h_{TE}}{R_{x}}},} & (88)\end{matrix}$

which is equal to the definition of the wave tilt, W. Finally, given thedesired Hankel crossover distance R_(x), the h_(TE) can be adjusted tomake the wave tilt of the incident ray match the complex Brewster angleat the Hankel crossover point 121. This can be accomplished by adjustingh_(p), Φ_(U), and/or h_(d).

These concepts may be better understood when discussed in the context ofan example of a guided surface waveguide probe. Referring to FIG. 14,shown is a graphical representation of an example of a guided surfacewaveguide probe 200 d including an upper charge terminal T₁ (e.g., asphere at height h_(T)) and a lower compensation terminal T₂ (e.g., adisk at height h_(d)) that are positioned along a vertical axis z thatis substantially normal to the plane presented by the lossy conductingmedium 203. During operation, charges Q₁ and Q₂ are imposed on thecharge and compensation terminals T₁ and T₂, respectively, depending onthe voltages applied to the terminals T₁ and T₂ at any given instant.

An AC source 212 acts as the excitation source for the charge terminalwhich is coupled to the guided surface waveguide probe 200 d through afeed network 209 comprising a coil 215 such as, e.g., a helical coil.The AC source 212 can be connected across a lower portion of the coil215 through a tap 227, as shown in FIG. 14, or can be inductivelycoupled to the coil 215 by way of a primary coil. The coil 215 can becoupled to a ground stake 218 at a first end and the charge terminal T₁at a second end. In some implementations, the connection to the chargeterminal T₁ can be adjusted using a tap 224 at the second end of thecoil 215. The compensation terminal T₂ is positioned above andsubstantially parallel with the lossy conducting medium 203 (e.g., theground or Earth), and energized through a tap 233 coupled to the coil215. An ammeter 236 located between the coil 215 and ground stake 218can be used to provide an indication of the magnitude of the currentflow (I₀) at the base of the guided surface waveguide probe.Alternatively, a current clamp may be used around the conductor coupledto the ground stake 218 to obtain an indication of the magnitude of thecurrent flow (I₀).

In the example of FIG. 14, the coil 215 is coupled to a ground stake 218at a first end and the charge terminal T₁ at a second end via a verticalfeed line conductor 221. In some implementations, the connection to thecharge terminal T₁ can be adjusted using a tap 224 at the second end ofthe coil 215 as shown in FIG. 14. The coil 215 can be energized at anoperating frequency by the AC source 212 through a tap 227 at a lowerportion of the coil 215. In other implementations, the AC source 212 canbe inductively coupled to the coil 215 through a primary coil. Thecompensation terminal T₂ is energized through a tap 233 coupled to thecoil 215. An ammeter 236 located between the coil 215 and ground stake218 can be used to provide an indication of the magnitude of the currentflow at the base of the guided surface waveguide probe 200 d.Alternatively, a current clamp may be used around the conductor coupledto the ground stake 218 to obtain an indication of the magnitude of thecurrent flow. The compensation terminal T₂ is positioned above andsubstantially parallel with the lossy conducting medium 203 (e.g., theground).

In the example of FIG. 14, the connection to the charge terminal T₁located on the coil 215 above the connection point of tap 233 for thecompensation terminal T₂. Such an adjustment allows an increased voltage(and thus a higher charge Q₁) to be applied to the upper charge terminalT₁. In other embodiments, the connection points for the charge terminalT₁ and the compensation terminal T₂ can be reversed. It is possible toadjust the total effective height (h_(TE)) of the guided surfacewaveguide probe 200 d to excite an electric field having a guidedsurface wave tilt at the Hankel crossover distance R_(x). The Hankelcrossover distance can also be found by equating the magnitudes ofequations (20b) and (21) for −jγρ, and solving for R_(x) as illustratedby FIG. 4. The index of refraction (n), the complex Brewster angle(θ_(i,B) and ψ_(i,B)), the wave tilt (|W|e^(jΨ)) and the complexeffective height (h_(eff)=h_(p)e^(jΦ)) can be determined as describedwith respect to Equations (41)-(44) above.

With the selected charge terminal T₁ configuration, a spherical diameter(or the effective spherical diameter) can be determined. For example, ifthe charge terminal T₁ is not configured as a sphere, then the terminalconfiguration may be modeled as a spherical capacitance having aneffective spherical diameter. The size of the charge terminal T₁ can bechosen to provide a sufficiently large surface for the charge Q₁ imposedon the terminals. In general, it is desirable to make the chargeterminal T₁ as large as practical. The size of the charge terminal T₁should be large enough to avoid ionization of the surrounding air, whichcan result in electrical discharge or sparking around the chargeterminal. To reduce the amount of bound charge on the charge terminalT₁, the desired elevation to provide free charge on the charge terminalT₁ for launching a guided surface wave should be at least 4-5 times theeffective spherical diameter above the lossy conductive medium (e.g.,the Earth). The compensation terminal T₂ can be used to adjust the totaleffective height (h_(TE)) of the guided surface waveguide probe 200 d toexcite an electric field having a guided surface wave tilt at R_(x). Thecompensation terminal T₂ can be positioned below the charge terminal T₁at h_(d)=h_(T)−h_(p), where h_(T) is the total physical height of thecharge terminal T₁. With the position of the compensation terminal T₂fixed and the phase delay Φ_(U) applied to the upper charge terminal T₁,the phase delay Φ_(L) applied to the lower compensation terminal T₂ canbe determined using the relationships of Equation (86), such that:

$\begin{matrix}{{\Phi_{U}\left( h_{d} \right)} = {{- {\beta \left( {h_{d} + y} \right)}} - {j\mspace{14mu} {{\ln \left( \frac{{R_{x} \times W} - {\left( {h_{p} + z} \right)e^{j{({{\beta \; h_{p}} + {\beta \; z} + \Phi_{L}})}}}}{\left( {h_{d} + y} \right)} \right)}.}}}} & (89)\end{matrix}$

In alternative embodiments, the compensation terminal T₂ can bepositioned at a height h_(d) where Im{(Φ_(L)}=0. This is graphicallyillustrated in FIG. 15A, which shows plots 172 and 175 of the imaginaryand real parts of Φ_(U), respectively. The compensation terminal T₂ ispositioned at a height h_(d) where Im{Φ_(U)}=0, as graphicallyillustrated in plot 172. At this fixed height, the coil phase Φ_(U) canbe determined from Re{Φ_(U)}, as graphically illustrated in plot 175.

With the AC source 212 coupled to the coil 215 (e.g., at the 50Ω pointto maximize coupling), the position of tap 233 may be adjusted forparallel resonance of the compensation terminal T₂ with at least aportion of the coil at the frequency of operation. FIG. 15B shows aschematic diagram of the general electrical hookup of FIG. 14 in whichV₁ is the voltage applied to the lower portion of the coil 215 from theAC source 212 through tap 227, V₂ is the voltage at tap 224 that issupplied to the upper charge terminal T₁, and V₃ is the voltage appliedto the lower compensation terminal T₂ through tap 233. The resistancesR_(p) and R_(d) represent the ground return resistances of the chargeterminal T₁ and compensation terminal T₂, respectively. The charge andcompensation terminals T₁ and T₂ may be configured as spheres,cylinders, toroids, rings, hoods, or any other combination of capacitivestructures. The size of the charge and compensation terminals T₁ and T₂can be chosen to provide a sufficiently large surface for the charges Q₁and Q₂ imposed on the terminals. In general, it is desirable to make thecharge terminal T₁ as large as practical. The size of the chargeterminal T₁ should be large enough to avoid ionization of thesurrounding air, which can result in electrical discharge or sparkingaround the charge terminal. The self-capacitance C_(p) and C_(d) of thecharge and compensation terminals T₁ and T₂ respectively, can bedetermined using, for example, equation (24).

As can be seen in FIG. 15B, a resonant circuit is formed by at least aportion of the inductance of the coil 215, the self-capacitance C_(d) ofthe compensation terminal T₂, and the ground return resistance R_(d)associated with the compensation terminal T₂. The parallel resonance canbe established by adjusting the voltage V₃ applied to the compensationterminal T₂ (e.g., by adjusting a tap 233 position on the coil 215) orby adjusting the height and/or size of the compensation terminal T₂ toadjust C_(d). The position of the coil tap 233 can be adjusted forparallel resonance, which will result in the ground current through theground stake 218 and through the ammeter 236 reaching a maximum point.After parallel resonance of the compensation terminal T₂ has beenestablished, the position of the tap 227 for the AC source 212 can beadjusted to the 500 point on the coil 215.

Voltage V₂ from the coil 215 can be applied to the charge terminal T₁,and the position of tap 224 can be adjusted such that the phase (Φ) ofthe total effective height (h_(TE)) approximately equals the angle ofthe guided surface wave tilt (W_(Rx)) at the Hankel crossover distance(R_(x)). The position of the coil tap 224 can be adjusted until thisoperating point is reached, which results in the ground current throughthe ammeter 236 increasing to a maximum. At this point, the resultantfields excited by the guided surface waveguide probe 200 d aresubstantially mode-matched to a guided surface waveguide mode on thesurface of the lossy conducting medium 203, resulting in the launchingof a guided surface wave along the surface of the lossy conductingmedium 203. This can be verified by measuring field strength along aradial extending from the guided surface waveguide probe 200.

Resonance of the circuit including the compensation terminal T₂ maychange with the attachment of the charge terminal T₁ and/or withadjustment of the voltage applied to the charge terminal T₁ through tap224. While adjusting the compensation terminal circuit for resonanceaids the subsequent adjustment of the charge terminal connection, it isnot necessary to establish the guided surface wave tilt (W_(Rx)) at theHankel crossover distance (R_(x)). The system may be further adjusted toimprove coupling by iteratively adjusting the position of the tap 227for the AC source 212 to be at the 500 point on the coil 215 andadjusting the position of tap 233 to maximize the ground current throughthe ammeter 236. Resonance of the circuit including the compensationterminal T₂ may drift as the positions of taps 227 and 233 are adjusted,or when other components are attached to the coil 215.

In other implementations, the voltage V₂ from the coil 215 can beapplied to the charge terminal T₁, and the position of tap 233 can beadjusted such that the phase (Φ) of the total effective height (h_(TE))approximately equals the angle (Ψ) of the guided surface wave tilt atR_(x). The position of the coil tap 224 can be adjusted until theoperating point is reached, resulting in the ground current through theammeter 236 substantially reaching a maximum. The resultant fields aresubstantially mode-matched to a guided surface waveguide mode on thesurface of the lossy conducting medium 203, and a guided surface wave islaunched along the surface of the lossy conducting medium 203. This canbe verified by measuring field strength along a radial extending fromthe guided surface waveguide probe 200. The system may be furtheradjusted to improve coupling by iteratively adjusting the position ofthe tap 227 for the AC source 212 to be at the 50Ω point on the coil 215and adjusting the position of tap 224 and/or 233 to maximize the groundcurrent through the ammeter 236.

Referring back to FIG. 12, operation of a guided surface waveguide probe200 may be controlled to adjust for variations in operational conditionsassociated with the guided surface waveguide probe 200. For example, aprobe control system 230 can be used to control the feed network 209and/or positioning of the charge terminal T₁ and/or compensationterminal T₂ to control the operation of the guided surface waveguideprobe 200. Operational conditions can include, but are not limited to,variations in the characteristics of the lossy conducting medium 203(e.g., conductivity σ and relative permittivity ε_(r)), variations infield strength and/or variations in loading of the guided surfacewaveguide probe 200. As can be seen from Equations (41)-(44), the indexof refraction (n), the complex Brewster angle (θ_(i,B) and ψ_(i,B)), thewave tilt (|W|e^(jΨ)) and the complex effective height(h_(eff)=h_(p)e^(jΦ)) can be affected by changes in soil conductivityand permittivity resulting from, e.g., weather conditions.

Equipment such as, e.g., conductivity measurement probes, permittivitysensors, ground parameter meters, field meters, current monitors and/orload receivers can be used to monitor for changes in the operationalconditions and provide information about current operational conditionsto the probe control system 230. The probe control system 230 can thenmake one or more adjustments to the guided surface waveguide probe 200to maintain specified operational conditions for the guided surfacewaveguide probe 200. For instance, as the moisture and temperature vary,the conductivity of the soil will also vary. Conductivity measurementprobes and/or permittivity sensors may be located at multiple locationsaround the guided surface waveguide probe 200. Generally, it would bedesirable to monitor the conductivity and/or permittivity at or aboutthe Hankel crossover distance R_(x) for the operational frequency.Conductivity measurement probes and/or permittivity sensors may belocated at multiple locations (e.g., in each quadrant) around the guidedsurface waveguide probe 200.

With reference then to FIG. 16, shown is an example of a guided surfacewaveguide probe 200 e that includes a charge terminal T₁ and a chargeterminal T₂ that are arranged along a vertical axis z. The guidedsurface waveguide probe 200 e is disposed above a lossy conductingmedium 203, which makes up Region 1. In addition, a second medium 206shares a boundary interface with the lossy conducting medium 203 andmakes up Region 2. The charge terminals T₁ and T₂ are positioned overthe lossy conducting medium 203. The charge terminal T₁ is positioned atheight H₁, and the charge terminal T₂ is positioned directly below T₁along the vertical axis z at height H₂, where H₂ is less than H₁. Theheight h of the transmission structure presented by the guided surfacewaveguide probe 200 e is h=H₁−H₂. The guided surface waveguide probe 200e includes a probe feed network 209 that couples an excitation source212 to the charge terminals T₁ and T₂.

The charge terminals T₁ and/or T₂ include a conductive mass that canhold an electrical charge, which may be sized to hold as much charge aspractically possible. The charge terminal T₁ has a self-capacitance C₁,and the charge terminal T₂ has a self-capacitance C₂, which can bedetermined using, for example, equation (24). By virtue of the placementof the charge terminal T₁ directly above the charge terminal T₂, amutual capacitance C_(M) is created between the charge terminals T₁ andT₂. Note that the charge terminals T₁ and T₂ need not be identical, buteach can have a separate size and shape, and can include differentconducting materials. Ultimately, the field strength of a guided surfacewave launched by a guided surface waveguide probe 200 e is directlyproportional to the quantity of charge on the terminal T₁. The charge Q₁is, in turn, proportional to the self-capacitance C₁ associated with thecharge terminal T₁ since Q₁=C₁V, where V is the voltage imposed on thecharge terminal T₁.

When properly adjusted to operate at a predefined operating frequency,the guided surface waveguide probe 200 e generates a guided surface wavealong the surface of the lossy conducting medium 203. The excitationsource 212 can generate electrical energy at the predefined frequencythat is applied to the guided surface waveguide probe 200 e to excitethe structure. When the electromagnetic fields generated by the guidedsurface waveguide probe 200 e are substantially mode-matched with thelossy conducting medium 203, the electromagnetic fields substantiallysynthesize a wave front incident at a complex Brewster angle thatresults in little or no reflection. Thus, the surface waveguide probe200 e does not produce a radiated wave, but launches a guided surfacetraveling wave along the surface of a lossy conducting medium 203. Theenergy from the excitation source 212 can be transmitted as Zennecksurface currents to one or more receivers that are located within aneffective transmission range of the guided surface waveguide probe 200e.

One can determine asymptotes of the radial Zenneck surface currentJ_(ρ)(ρ) on the surface of the lossy conducting medium 203 to be J₁(ρ)close-in and J₂(ρ) far-out, where

$\begin{matrix}{{{{{Close}\text{-}{in}\mspace{14mu} \left( {\rho < {\lambda \text{/}8}} \right)\text{:}\mspace{14mu} {J_{\rho}(\rho)}} \sim J_{1}} = {\frac{I_{1} + I_{2}}{2\; \pi \; \rho} + \frac{{E_{\rho}^{QS}\left( Q_{1} \right)} + {E_{\rho}^{QS}\left( Q_{2} \right)}}{Z_{\rho}}}},{and}} & (90) \\{{{{Far}\text{-}{out}\mspace{14mu} \left( {\rho {\lambda \text{/}8}} \right)\text{:}\mspace{14mu} {J_{\rho}(\rho)}} \sim J_{2}} = {\frac{j\; \gamma \; \omega \; Q_{1}}{4} \times \sqrt{\frac{2\; \gamma}{\pi}} \times {\frac{e^{{- {({\alpha + {j\; \beta}})}}\rho}}{\sqrt{\rho}}.}}} & (91)\end{matrix}$

where I₁ is the conduction current feeding the charge Q₁ on the firstcharge terminal T₁, and I₂ is the conduction current feeding the chargeQ₂ on the second charge terminal T₂. The charge Q₁ on the upper chargeterminal T₁ is determined by Q₁=C₁V₁, where C₁ is the isolatedcapacitance of the charge terminal T₁. Note that there is a thirdcomponent to J₁ set forth above given by (E_(ρ) ^(Q) ¹ )/Z_(ρ), whichfollows from the Leontovich boundary condition and is the radial currentcontribution in the lossy conducting medium 203 pumped by thequasi-static field of the elevated oscillating charge on the firstcharge terminal Q₁. The quantity Z_(ρ)=jωμ₀/γ_(e) is the radialimpedance of the lossy conducting medium, whereγ_(e)=(jωμ₁σ₁−ω²μ₁ε₁)^(1/2).

The asymptotes representing the radial current close-in and far-out asset forth by equations (90) and (91) are complex quantities. Accordingto various embodiments, a physical surface current J(ρ), is synthesizedto match as close as possible the current asymptotes in magnitude andphase. That is to say close-in, |J(ρ)| is to be tangent to |J₁|, andfar-out |J(ρ)| is to be tangent to |J₂|. Also, according to the variousembodiments, the phase of J(p) should transition from the phase of J₁close-in to the phase of J₂ far-out.

In order to match the guided surface wave mode at the site oftransmission to launch a guided surface wave, the phase of the surfacecurrent |J₂|far-out should differ from the phase of the surface current|J₁| close-in by the propagation phase corresponding to e^(−jβ(ρ) ²^(−ρ) ¹ ⁾ plus a constant of approximately 45 degrees or 225 degrees.This is because there are two roots for √{square root over (γ)}, onenear π/4 and one near 5π/4. The properly adjusted synthetic radialsurface current is

$\begin{matrix}{{J_{\rho}\left( {\rho,\varphi,0} \right)} = {\frac{I_{O}\gamma}{4}\mspace{14mu} {{H_{1}^{(2)}\left( {{- j}\; {\gamma\rho}} \right)}.}}} & (92)\end{matrix}$

Note that this is consistent with equation (17). By Maxwell's equations,such a J(ρ) surface current automatically creates fields that conform to

$\begin{matrix}{{H_{\varphi} = {\frac{{- \gamma}\; I_{o}}{4}e^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- j}\; \gamma \; \rho} \right)}}},} & (93) \\{{E_{\rho} = {\frac{{- \gamma}\; I_{o}}{4}\left( \frac{u_{2}}{j\; \omega \; ɛ_{o}} \right)e^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- j}\; \gamma \; \rho} \right)}}},{and}} & (94) \\{E_{z} = {\frac{{- \gamma}\; I_{O}}{4}\left( \frac{- \gamma}{\omega \; ɛ_{o}} \right)e^{{- u_{2}}z}{{H_{0}^{(2)}\left( {{- j}\; \gamma \; \rho} \right)}.}}} & (95)\end{matrix}$

Thus, the difference in phase between the surface current |J₂| far-outand the surface current |J₁| close-in for the guided surface wave modethat is to be matched is due to the characteristics of the Hankelfunctions in equations (93)-(95), which are consistent with equations(1)-(3). It is of significance to recognize that the fields expressed byequations (1)-(6) and (17) and equations (92)-(95) have the nature of atransmission line mode bound to a lossy interface, not radiation fieldsthat are associated with groundwave propagation.

In order to obtain the appropriate voltage magnitudes and phases for agiven design of a guided surface waveguide probe 200 e at a givenlocation, an iterative approach may be used. Specifically, analysis maybe performed of a given excitation and configuration of a guided surfacewaveguide probe 200 e taking into account the feed currents to theterminals T₁ and T₂, the charges on the charge terminals T₁ and T₂, andtheir images in the lossy conducting medium 203 in order to determinethe radial surface current density generated. This process may beperformed iteratively until an optimal configuration and excitation fora given guided surface waveguide probe 200 e is determined based ondesired parameters. To aid in determining whether a given guided surfacewaveguide probe 200 e is operating at an optimal level, a guided fieldstrength curve 103 (FIG. 1) may be generated using equations (1)-(12)based on values for the conductivity of Region 1 (σ₁) and thepermittivity of Region 1 (ε₁) at the location of the guided surfacewaveguide probe 200 e. Such a guided field strength curve 103 canprovide a benchmark for operation such that measured field strengths canbe compared with the magnitudes indicated by the guided field strengthcurve 103 to determine if optimal transmission has been achieved.

In order to arrive at an optimized condition, various parametersassociated with the guided surface waveguide probe 200 e may beadjusted. One parameter that may be varied to adjust the guided surfacewaveguide probe 200 e is the height of one or both of the chargeterminals T₁ and/or T₂ relative to the surface of the lossy conductingmedium 203. In addition, the distance or spacing between the chargeterminals T₁ and T₂ may also be adjusted. In doing so, one may minimizeor otherwise alter the mutual capacitance C_(M) or any boundcapacitances between the charge terminals T₁ and T₂ and the lossyconducting medium 203 as can be appreciated. The size of the respectivecharge terminals T₁ and/or T₂ can also be adjusted. By changing the sizeof the charge terminals T₁ and/or T₂, one will alter the respectiveself-capacitances C₁ and/or C₂, and the mutual capacitance C_(M) as canbe appreciated.

Still further, another parameter that can be adjusted is the feednetwork 209 associated with the guided surface waveguide probe 200 e.This may be accomplished by adjusting the size of the inductive and/orcapacitive reactances that make up the feed network 209. For example,where such inductive reactances comprise coils, the number of turns onsuch coils may be adjusted. Ultimately, the adjustments to the feednetwork 209 can be made to alter the electrical length of the feednetwork 209, thereby affecting the voltage magnitudes and phases on thecharge terminals T₁ and T₂.

Note that the iterations of transmission performed by making the variousadjustments may be implemented by using computer models or by adjustingphysical structures as can be appreciated. By making the aboveadjustments, one can create corresponding “close-in” surface current J₁and “far-out” surface current J₂ that approximate the same currents J(ρ)of the guided surface wave mode specified in Equations (90) and (91) setforth above. In doing so, the resulting electromagnetic fields would besubstantially or approximately mode-matched to a guided surface wavemode on the surface of the lossy conducting medium 203.

While not shown in the example of FIG. 16, operation of the guidedsurface waveguide probe 200 e may be controlled to adjust for variationsin operational conditions associated with the guided surface waveguideprobe 200. For example, a probe control system 230 shown in FIG. 12 canbe used to control the feed network 209 and/or positioning and/or sizeof the charge terminals T₁ and/or T₂ to control the operation of theguided surface waveguide probe 200 e. Operational conditions caninclude, but are not limited to, variations in the characteristics ofthe lossy conducting medium 203 (e.g., conductivity σ and relativepermittivity ε_(r)), variations in field strength and/or variations inloading of the guided surface waveguide probe 200 e.

Referring now to FIG. 17, shown is an example of the guided surfacewaveguide probe 200 e of FIG. 16, denoted herein as guided surfacewaveguide probe 200 f. The guided surface waveguide probe 200 f includesthe charge terminals T₁ and T₂ that are positioned along a vertical axisz that is substantially normal to the plane presented by the lossyconducting medium 203 (e.g., the Earth). The second medium 206 is abovethe lossy conducting medium 203. The charge terminal T₁ has aself-capacitance C₁, and the charge terminal T₂ has a self-capacitanceC₂. During operation, charges Q₁ and Q₂ are imposed on the chargeterminals T₁ and T₂, respectively, depending on the voltages applied tothe charge terminals T₁ and T₂ at any given instant. A mutualcapacitance C_(M) may exist between the charge terminals T₁ and T₂depending on the distance there between. In addition, bound capacitancesmay exist between the respective charge terminals T₁ and T₂ and thelossy conducting medium 203 depending on the heights of the respectivecharge terminals T₁ and T₂ with respect to the lossy conducting medium203.

The guided surface waveguide probe 200 f includes a feed network 209that comprises an inductive impedance comprising a coil L_(1a) having apair of leads that are coupled to respective ones of the chargeterminals T₁ and T₂. In one embodiment, the coil L_(1a) is specified tohave an electrical length that is one-half (½) of the wavelength at theoperating frequency of the guided surface waveguide probe 200 f.

While the electrical length of the coil L_(1a) is specified asapproximately one-half (½) the wavelength at the operating frequency, itis understood that the coil L_(1a) may be specified with an electricallength at other values. According to one embodiment, the fact that thecoil L_(1a) has an electrical length of approximately one-half thewavelength at the operating frequency provides for an advantage in thata maximum voltage differential is created on the charge terminals T₁ andT₂. Nonetheless, the length or diameter of the coil L_(1a) may beincreased or decreased when adjusting the guided surface waveguide probe200 f to obtain optimal excitation of a guided surface wave mode.Adjustment of the coil length may be provided by taps located at one orboth ends of the coil. In other embodiments, it may be the case that theinductive impedance is specified to have an electrical length that issignificantly less than or greater than ½ the wavelength at theoperating frequency of the guided surface waveguide probe 200 f.

The excitation source 212 can be coupled to the feed network 209 by wayof magnetic coupling. Specifically, the excitation source 212 is coupledto a coil L_(P) that is inductively coupled to the coil L_(1a). This maybe done by link coupling, a tapped coil, a variable reactance, or othercoupling approach as can be appreciated. To this end, the coil L_(P)acts as a primary, and the coil L_(1a) acts as a secondary as can beappreciated.

In order to adjust the guided surface waveguide probe 200 f for thetransmission of a desired guided surface wave, the heights of therespective charge terminals T₁ and T₂ may be altered with respect to thelossy conducting medium 203 and with respect to each other. Also, thesizes of the charge terminals T₁ and T₂ may be altered. In addition, thesize of the coil L_(1a) may be altered by adding or eliminating turns orby changing some other dimension of the coil L_(1a). The coil L_(1a) canalso include one or more taps for adjusting the electrical length asshown in FIG. 17. The position of a tap connected to either chargeterminal T₁ or T₂ can also be adjusted.

Referring next to FIGS. 18A, 18B, 18C and 19, shown are examples ofgeneralized receive circuits for using the surface-guided waves inwireless power delivery systems. FIGS. 18A and 18B-18C include a linearprobe 303 and a tuned resonator 306, respectively. FIG. 19 is a magneticcoil 309 according to various embodiments of the present disclosure.According to various embodiments, each one of the linear probe 303, thetuned resonator 306, and the magnetic coil 309 may be employed toreceive power transmitted in the form of a guided surface wave on thesurface of a lossy conducting medium 203 according to variousembodiments. As mentioned above, in one embodiment the lossy conductingmedium 203 comprises a terrestrial medium (or Earth).

With specific reference to FIG. 18A, the open-circuit terminal voltageat the output terminals 312 of the linear probe 303 depends upon theeffective height of the linear probe 303. To this end, the terminalpoint voltage may be calculated as

V _(T)=∫₀ ^(h) ^(e) E _(inc) ·dl,  (96)

where E_(inc) is the strength of the incident electric field induced onthe linear probe 303 in Volts per meter, dl is an element of integrationalong the direction of the linear probe 303, and h_(e) is the effectiveheight of the linear probe 303. An electrical load 315 is coupled to theoutput terminals 312 through an impedance matching network 318.

When the linear probe 303 is subjected to a guided surface wave asdescribed above, a voltage is developed across the output terminals 312that may be applied to the electrical load 315 through a conjugateimpedance matching network 318 as the case may be. In order tofacilitate the flow of power to the electrical load 315, the electricalload 315 should be substantially impedance matched to the linear probe303 as will be described below.

Referring to FIG. 18B, a ground current excited coil 306 a possessing aphase shift equal to the wave tilt of the guided surface wave includes acharge terminal T_(R) that is elevated (or suspended) above the lossyconducting medium 203. The charge terminal T_(R) has a self-capacitanceC_(R). In addition, there may also be a bound capacitance (not shown)between the charge terminal T_(R) and the lossy conducting medium 203depending on the height of the charge terminal T_(R) above the lossyconducting medium 203. The bound capacitance should preferably beminimized as much as is practicable, although this may not be entirelynecessary in every instance.

The tuned resonator 306 a also includes a receiver network comprising acoil L_(R) having a phase shift Φ. One end of the coil L_(R) is coupledto the charge terminal T_(R), and the other end of the coil L_(R) iscoupled to the lossy conducting medium 203. The receiver network caninclude a vertical supply line conductor that couples the coil L_(R) tothe charge terminal T_(R). To this end, the coil L_(R) (which may alsobe referred to as tuned resonator L_(R)-C_(R)) comprises aseries-adjusted resonator as the charge terminal C_(R) and the coilL_(R) are situated in series. The phase delay of the coil L_(R) can beadjusted by changing the size and/or height of the charge terminalT_(R), and/or adjusting the size of the coil L_(R) so that the phase Φof the structure is made substantially equal to the angle of the wavetilt Ψ. The phase delay of the vertical supply line can also be adjustedby, e.g., changing length of the conductor.

For example, the reactance presented by the self-capacitance C_(R) iscalculated as 1/jωC_(R). Note that the total capacitance of thestructure 306 a may also include capacitance between the charge terminalT_(R) and the lossy conducting medium 203, where the total capacitanceof the structure 306 a may be calculated from both the self-capacitanceC_(R) and any bound capacitance as can be appreciated. According to oneembodiment, the charge terminal T_(R) may be raised to a height so as tosubstantially reduce or eliminate any bound capacitance. The existenceof a bound capacitance may be determined from capacitance measurementsbetween the charge terminal T_(R) and the lossy conducting medium 203 aspreviously discussed.

The inductive reactance presented by a discrete-element coil L_(R) maybe calculated as jωL, where L is the lumped-element inductance of thecoil L_(R). If the coil L_(R) is a distributed element, its equivalentterminal-point inductive reactance may be determined by conventionalapproaches. To tune the structure 306 a, one would make adjustments sothat the phase delay is equal to the wave tilt for the purpose ofmode-matching to the surface waveguide at the frequency of operation.Under this condition, the receiving structure may be considered to be“mode-matched” with the surface waveguide. A transformer link around thestructure and/or an impedance matching network 324 may be insertedbetween the probe and the electrical load 327 in order to couple powerto the load. Inserting the impedance matching network 324 between theprobe terminals 321 and the electrical load 327 can effect aconjugate-match condition for maximum power transfer to the electricalload 327.

When placed in the presence of surface currents at the operatingfrequencies power will be delivered from the surface guided wave to theelectrical load 327. To this end, an electrical load 327 may be coupledto the structure 306 a by way of magnetic coupling, capacitive coupling,or conductive (direct tap) coupling. The elements of the couplingnetwork may be lumped components or distributed elements as can beappreciated.

In the embodiment shown in FIG. 18B, magnetic coupling is employed wherea coil LS is positioned as a secondary relative to the coil L_(R) thatacts as a transformer primary. The coil Ls may be link-coupled to thecoil L_(R) by geometrically winding it around the same core structureand adjusting the coupled magnetic flux as can be appreciated. Inaddition, while the receiving structure 306 a comprises a series-tunedresonator, a parallel-tuned resonator or even a distributed-elementresonator of the appropriate phase delay may also be used.

While a receiving structure immersed in an electromagnetic field maycouple energy from the field, it can be appreciated thatpolarization-matched structures work best by maximizing the coupling,and conventional rules for probe-coupling to waveguide modes should beobserved. For example, a TE₂₀ (transverse electric mode) waveguide probemay be optimal for extracting energy from a conventional waveguideexcited in the TE₂₀ mode. Similarly, in these cases, a mode-matched andphase-matched receiving structure can be optimized for coupling powerfrom a surface-guided wave. The guided surface wave excited by a guidedsurface waveguide probe 200 on the surface of the lossy conductingmedium 203 can be considered a waveguide mode of an open waveguide.Excluding waveguide losses, the source energy can be completelyrecovered. Useful receiving structures may be E-field coupled, H-fieldcoupled, or surface-current excited.

The receiving structure can be adjusted to increase or maximize couplingwith the guided surface wave based upon the local characteristics of thelossy conducting medium 203 in the vicinity of the receiving structure.To accomplish this, the phase delay (Φ) of the receiving structure canbe adjusted to match the angle (Ψ) of the wave tilt of the surfacetraveling wave at the receiving structure. If configured appropriately,the receiving structure may then be tuned for resonance with respect tothe perfectly conducting image ground plane at complex depth z=−d/2.

For example, consider a receiving structure comprising the tunedresonator 306 a of FIG. 18B, including a coil L_(R) and a verticalsupply line connected between the coil L_(R) and a charge terminalT_(R). With the charge terminal T_(R) positioned at a defined heightabove the lossy conducting medium 203, the total phase shift Φ of thecoil L_(R) and vertical supply line can be matched with the angle (Ψ) ofthe wave tilt at the location of the tuned resonator 306 a. FromEquation (22), it can be seen that the wave tilt asymptotically passesto

$\begin{matrix}{{W = {{{W}e^{j\; \Psi}} = {\frac{E_{\rho}}{E_{z}}\underset{\mspace{20mu} {\rho\rightarrow\infty}\mspace{11mu}}{\rightarrow}\frac{1}{\sqrt{ɛ_{r} - {j\frac{\sigma_{1}}{{\omega ɛ}_{o}}}}}}}},} & (97)\end{matrix}$

where ε_(r) comprises the relative permittivity and σ₁ is theconductivity of the lossy conducting medium 203 at the location of thereceiving structure, ε₀ is the permittivity of free space, and ω=2πf,where f is the frequency of excitation. Thus, the wave tilt angle (Ψ)can be determined from Equation (97).

The total phase shift (Φ=θ_(c)+θ_(y)) of the tuned resonator 306 aincludes both the phase delay (θ_(c)) through the coil L_(R) and thephase delay of the vertical supply line (θ_(y)). The spatial phase delayalong the conductor length l_(w) of the vertical supply line can begiven by θ_(y)=β_(w)l_(w), where β_(w) is the propagation phase constantfor the vertical supply line conductor. The phase delay due to the coil(or helical delay line) is θ_(c)=β_(p)l_(C), with a physical length ofl_(C) and a propagation factor of

$\begin{matrix}{{\beta_{p} = {\frac{2\pi}{\lambda_{p}} = \frac{2\pi}{V_{f}\lambda_{o}}}},} & (98)\end{matrix}$

where V_(f) is the velocity factor on the structure, λ₀ is thewavelength at the supplied frequency, and λ_(p) is the propagationwavelength resulting from the velocity factor V_(f). One or both of thephase delays (θ_(c)+θ_(y)) can be adjusted to match the phase shift Φ tothe angle (Ψ) of the wave tilt. For example, a tap position may beadjusted on the coil L_(R) of FIG. 18B to adjust the coil phase delay(θ_(c)) to match the total phase shift to the wave tilt angle (Φ=Ψ). Forexample, a portion of the coil can be bypassed by the tap connection asillustrated in FIG. 18B. The vertical supply line conductor can also beconnected to the coil L_(R) via a tap, whose position on the coil may beadjusted to match the total phase shift to the angle of the wave tilt.

Once the phase delay (Φ) of the tuned resonator 306 a has been adjusted,the impedance of the charge terminal T_(R) can then be adjusted to tuneto resonance with respect to the perfectly conducting image ground planeat complex depth z=−d/2. This can be accomplished by adjusting thecapacitance of the charge terminal T₁ without changing the travelingwave phase delays of the coil L_(R) and vertical supply line. Theadjustments are similar to those described with respect to FIGS. 9A and9B.

The impedance seen “looking down” into the lossy conducting medium 203to the complex image plane is given by:

Z _(in) =R _(in) +jX _(in) =Z ₀ tan h(jβ ₀(d/2)),  (99)

where β₀=√{square root over (ωμ₀ε₀)}. For vertically polarized sourcesover the Earth, the depth of the complex image plane can be given by:

d/2≈1/√{square root over (jωμ ₁σ₁−ω²μ₁ε₁)},  (100)

where μ₁ is the permeability of the lossy conducting medium 203 andε₁=ε_(r)ε₀.

At the base of the tuned resonator 306 a, the impedance seen “lookingup” into the receiving structure is Z_(↑)=Z_(base) as illustrated inFIG. 9A. With a terminal impedance of:

$\begin{matrix}{{Z_{R} = \frac{1}{j\; \omega \; C_{R}}},} & (101)\end{matrix}$

where C_(R) is the self-capacitance of the charge terminal T_(R), theimpedance seen “looking up” into the vertical supply line conductor ofthe tuned resonator 306 a is given by:

$\begin{matrix}{{Z_{2} = {{Z_{W}\frac{Z_{R} + {Z_{w}{\tanh \left( {j\; \beta_{w}h_{w}} \right)}}}{Z_{w} + {Z_{R}{\tanh \left( {j\; \beta_{w}h_{w}} \right)}}}} = {Z_{W}\frac{Z_{R} + {Z_{w}{\tanh \left( {j\; \theta_{y}} \right)}}}{Z_{w} + {Z_{R}{\tanh \left( {j\; \theta_{y}} \right)}}}}}},} & (102)\end{matrix}$

and the impedance seen “looking up” into the coil L_(R) of the tunedresonator 306 a is given by:

$\begin{matrix}{Z_{base} = {{R_{base} + {jX}_{base}} = {{Z_{R}\frac{Z_{2} + {Z_{R}{\tanh \left( {j\; \beta_{p}H} \right)}}}{Z_{R} + {Z_{2}{\tanh \left( {j\; \beta_{p}H} \right)}}}} = {Z_{c}{\frac{Z_{2} + {Z_{R}{\tanh \left( {j\; \theta_{c}} \right)}}}{Z_{R} + {Z_{2}{\tanh \left( {j\; \theta_{c}} \right)}}}.}}}}} & (103)\end{matrix}$

By matching the reactive component (X_(in)) seen “looking down” into thelossy conducting medium 203 with the reactive component (X_(base)) seen“looking up” into the tuned resonator 306 a, the coupling into theguided surface waveguide mode may be maximized.

Referring next to FIG. 18C, shown is an example of a tuned resonator 306b that does not include a charge terminal T_(R) at the top of thereceiving structure. In this embodiment, the tuned resonator 306 b doesnot include a vertical supply line coupled between the coil L_(R) andthe charge terminal T_(R). Thus, the total phase shift (Φ) of the tunedresonator 306 b includes only the phase delay (θ_(c)) through the coilL_(R). As with the tuned resonator 306 a of FIG. 18B, the coil phasedelay θ_(c) can be adjusted to match the angle (Ψ) of the wave tiltdetermined from Equation (97), which results in Φ=Ψ. While powerextraction is possible with the receiving structure coupled into thesurface waveguide mode, it is difficult to adjust the receivingstructure to maximize coupling with the guided surface wave without thevariable reactive load provided by the charge terminal T_(R).

Referring to FIG. 18D, shown is a flow chart 180 illustrating an exampleof adjusting a receiving structure to substantially mode-match to aguided surface waveguide mode on the surface of the lossy conductingmedium 203. Beginning with 181, if the receiving structure includes acharge terminal T_(R) (e.g., of the tuned resonator 306 a of FIG. 18B),then the charge terminal T_(R) is positioned at a defined height above alossy conducting medium 203 at 184. As the surface guided wave has beenestablished by a guided surface waveguide probe 200, the physical height(h_(p)) of the charge terminal T_(R) may be below that of the effectiveheight. The physical height may be selected to reduce or minimize thebound charge on the charge terminal T_(R) (e.g., four times thespherical diameter of the charge terminal). If the receiving structuredoes not include a charge terminal T_(R) (e.g., of the tuned resonator306 b of FIG. 18C), then the flow proceeds to 187.

At 187, the electrical phase delay Φ of the receiving structure ismatched to the complex wave tilt angle Ψ defined by the localcharacteristics of the lossy conducting medium 203. The phase delay(θ_(c)) of the helical coil and/or the phase delay (θ_(y)) of thevertical supply line can be adjusted to make Φ equal to the angle (Ψ) ofthe wave tilt (W). The angle (Ψ) of the wave tilt can be determined fromEquation (86). The electrical phase Φ can then be matched to the angleof the wave tilt. For example, the electrical phase delay Φ=θ_(c)+θ_(y)can be adjusted by varying the geometrical parameters of the coil L_(R)and/or the length (or height) of the vertical supply line conductor.

Next at 190, the load impedance of the charge terminal T_(R) can betuned to resonate the equivalent image plane model of the tunedresonator 306 a. The depth (d/2) of the conducting image ground plane139 (FIG. 9A) below the receiving structure can be determined usingEquation (100) and the values of the lossy conducting medium 203 (e.g.,the Earth) at the receiving structure, which can be locally measured.Using that complex depth, the phase shift (θ_(d)) between the imageground plane 139 and the physical boundary 136 (FIG. 9A) of the lossyconducting medium 203 can be determined using θ_(d)=β₀d/2. The impedance(Z_(in)) as seen “looking down” into the lossy conducting medium 203 canthen be determined using Equation (99). This resonance relationship canbe considered to maximize coupling with the guided surface waves.

Based upon the adjusted parameters of the coil L_(R) and the length ofthe vertical supply line conductor, the velocity factor, phase delay,and impedance of the coil L_(R) and vertical supply line can bedetermined. In addition, the self-capacitance (C_(R)) of the chargeterminal T_(R) can be determined using, e.g., Equation (24). Thepropagation factor (β_(p)) of the coil L_(R) can be determined usingEquation (98), and the propagation phase constant (β_(w)) for thevertical supply line can be determined using Equation (49). Using theself-capacitance and the determined values of the coil L_(R) andvertical supply line, the impedance (Z_(base)) of the tuned resonator306 a as seen “looking up” into the coil L_(R) can be determined usingEquations (101), (102), and (103).

The equivalent image plane model of FIG. 9A also applies to the tunedresonator 306 a of FIG. 18B. The tuned resonator 306 a can be tuned toresonance with respect to the complex image plane by adjusting the loadimpedance Z_(R) of the charge terminal T_(R) such that the reactancecomponent X_(base) Of Z_(base) cancels out the reactance component ofX_(in) of Z_(in), or X_(base) X_(in)=0. Thus, the impedance at thephysical boundary 136 (FIG. 9A) “looking up” into the coil of the tunedresonator 306 a is the conjugate of the impedance at the physicalboundary 136 “looking down” into the lossy conducting medium 203. Theload impedance Z_(R) can be adjusted by varying the capacitance (C_(R))of the charge terminal T_(R) without changing the electrical phase delayΦ=θ_(c)+θ_(y) seen by the charge terminal T_(R). An iterative approachmay be taken to tune the load impedance Z_(R) for resonance of theequivalent image plane model with respect to the conducting image groundplane 139. In this way, the coupling of the electric field to a guidedsurface waveguide mode along the surface of the lossy conducting medium203 (e.g., Earth) can be improved and/or maximized.

Referring to FIG. 19, the magnetic coil 309 comprises a receive circuitthat is coupled through an impedance matching network 333 to anelectrical load 336. In order to facilitate reception and/or extractionof electrical power from a guided surface wave, the magnetic coil 309may be positioned so that the magnetic flux of the guided surface wave,H_(φ), passes through the magnetic coil 309, thereby inducing a currentin the magnetic coil 309 and producing a terminal point voltage at itsoutput terminals 330. The magnetic flux of the guided surface wavecoupled to a single turn coil is expressed by

=∫∫_(A) _(cs) μ_(r)μ₀ {right arrow over (H)}·{circumflex over(n)}dA  (104)

where

is the coupled magnetic flux, μ_(r) is the effective relativepermeability of the core of the magnetic coil 309, μ₀ is thepermeability of free space, {right arrow over (H)} is the incidentmagnetic field strength vector, {circumflex over (n)} is a unit vectornormal to the cross-sectional area of the turns, and A_(CS) is the areaenclosed by each loop. For an N-turn magnetic coil 309 oriented formaximum coupling to an incident magnetic field that is uniform over thecross-sectional area of the magnetic coil 309, the open-circuit inducedvoltage appearing at the output terminals 330 of the magnetic coil 309is

$\begin{matrix}{{V = {{{- N}\frac{d\; \mathcal{F}}{dt}} \approx {{- j}\; {\omega\mu}_{r}\mu_{0}{NHA}_{CS}}}},} & (105)\end{matrix}$

where the variables are defined above. The magnetic coil 309 may betuned to the guided surface wave frequency either as a distributedresonator or with an external capacitor across its output terminals 330,as the case may be, and then impedance-matched to an external electricalload 336 through a conjugate impedance matching network 333.

Assuming that the resulting circuit presented by the magnetic coil 309and the electrical load 336 are properly adjusted and conjugateimpedance matched, via impedance matching network 333, then the currentinduced in the magnetic coil 309 may be employed to optimally power theelectrical load 336. The receive circuit presented by the magnetic coil309 provides an advantage in that it does not have to be physicallyconnected to the ground.

With reference to FIGS. 18A, 18B, 18C and 19, the receive circuitspresented by the linear probe 303, the mode-matched structure 306, andthe magnetic coil 309 each facilitate receiving electrical powertransmitted from any one of the embodiments of guided surface waveguideprobes 200 described above. To this end, the energy received may be usedto supply power to an electrical load 315/327/336 via a conjugatematching network as can be appreciated. This contrasts with the signalsthat may be received in a receiver that were transmitted in the form ofa radiated electromagnetic field. Such signals have very low availablepower, and receivers of such signals do not load the transmitters.

It is also characteristic of the present guided surface waves generatedusing the guided surface waveguide probes 200 described above that thereceive circuits presented by the linear probe 303, the mode-matchedstructure 306, and the magnetic coil 309 will load the excitation source212 (e.g., FIGS. 3, 12 and 16) that is applied to the guided surfacewaveguide probe 200, thereby generating the guided surface wave to whichsuch receive circuits are subjected. This reflects the fact that theguided surface wave generated by a given guided surface waveguide probe200 described above comprises a transmission line mode. By way ofcontrast, a power source that drives a radiating antenna that generatesa radiated electromagnetic wave is not loaded by the receivers,regardless of the number of receivers employed.

Thus, together one or more guided surface waveguide probes 200 and oneor more receive circuits in the form of the linear probe 303, the tunedmode-matched structure 306, and/or the magnetic coil 309 can make up awireless distribution system. Given that the distance of transmission ofa guided surface wave using a guided surface waveguide probe 200 as setforth above depends upon the frequency, it is possible that wirelesspower distribution can be achieved across wide areas and even globally.

The conventional wireless-power transmission/distribution systemsextensively investigated today include “energy harvesting” fromradiation fields and also sensor coupling to inductive or reactivenear-fields. In contrast, the present wireless-power system does notwaste power in the form of radiation which, if not intercepted, is lostforever. Nor is the presently disclosed wireless-power system limited toextremely short ranges as with conventional mutual-reactance couplednear-field systems. The wireless-power system disclosed hereinprobe-couples to the novel surface-guided transmission line mode, whichis equivalent to delivering power to a load by a wave-guide or a loaddirectly wired to the distant power generator. Not counting the powerrequired to maintain transmission field strength plus that dissipated inthe surface waveguide, which at extremely low frequencies isinsignificant relative to the transmission losses in conventionalhigh-tension power lines at 60 Hz, all of the generator power goes onlyto the desired electrical load. When the electrical load demand isterminated, the source power generation is relatively idle.

Referring next to FIGS. 20A-E, shown are examples of various schematicsymbols that are used with reference to the discussion that follows.With specific reference to FIG. 20A, shown is a symbol that representsany one of the guided surface waveguide probes 200 a, 200 b, 200 c, 200e, 200 d, or 200 f, or any variations thereof. In the following drawingsand discussion, a depiction of this symbol will be referred to as aguided surface waveguide probe P. For the sake of simplicity in thefollowing discussion, any reference to the guided surface waveguideprobe P is a reference to any one of the guided surface waveguide probes200 a, 200 b, 200 c, 200 e, 200 d, or 200 f, or variations thereof.

Similarly, with reference to FIG. 20B, shown is a symbol that representsa guided surface wave receive structure that may comprise any one of thelinear probe 303 (FIG. 18A), the tuned resonator 306 (FIGS. 18B-18C), orthe magnetic coil 309 (FIG. 19). In the following drawings anddiscussion, a depiction of this symbol will be referred to as a guidedsurface wave receive structure R. For the sake of simplicity in thefollowing discussion, any reference to the guided surface wave receivestructure R is a reference to any one of the linear probe 303, the tunedresonator 306, or the magnetic coil 309 or variations thereof.

Further, with reference to FIG. 20C, shown is a symbol that specificallyrepresents the linear probe 303 (FIG. 18A). In the following drawingsand discussion, a depiction of this symbol will be referred to as aguided surface wave receive structure R_(P). For the sake of simplicityin the following discussion, any reference to the guided surface wavereceive structure R_(P) is a reference to the linear probe 303 orvariations thereof.

Further, with reference to FIG. 20D, shown is a symbol that specificallyrepresents the tuned resonator 306 (FIGS. 18B-18C). In the followingdrawings and discussion, a depiction of this symbol will be referred toas a guided surface wave receive structure RR. For the sake ofsimplicity in the following discussion, any reference to the guidedsurface wave receive structure RR is a reference to the tuned resonator306 or variations thereof.

Further, with reference to FIG. 20E, shown is a symbol that specificallyrepresents the magnetic coil 309 (FIG. 19). In the following drawingsand discussion, a depiction of this symbol will be referred to as aguided surface wave receive structure R_(M). For the sake of simplicityin the following discussion, any reference to the guided surface wavereceive structure R_(M) is a reference to the magnetic coil 309 orvariations thereof.

With reference to FIG. 21, shown is an example of a modulatedtransmission system such as an Amplitude Modulation (AM) transmissionsystem 400 according to one embodiment of the present disclosure. The AMtransmission system 400 includes an AM transmitter 403, a matchingnetwork 406, a guided surface waveguide probe P, and potentially othercomponents.

In the following discussion, the various embodiments are discussed withreference to Amplitude Modulation (AM). However, it is understood thatmany other types of modulation may be employed. Consequently, asdescribed herein, an AM transmitter provides an example of the varioustypes of transmitters that may be employed. To this end, other types ofmodulation that may be employed beyond AM transmission may comprise, forexample, frequency modulation, frequency-shift keying, packetmodulation, and other modulation techniques. Accordingly, wherereference is made to an AM transmitter or another other component oraspect involved in AM transmission, it is understood that the sameprinciples apply to other modulation techniques and that such othermodulation equipment and techniques may be substituted for AMtransmission where appropriate. Thus, as contemplated herein, terms suchas a modulation transmitter, modulated signal, and the like refer to alltypes of modulation and not merely amplitude modulation. Accordingly, asdescribed herein, an amplitude modulation transmission system is merelyan example of a modulated transmission system, an AM transmitter ismerely an example of a modulation transmitter or an AM signal is merelyan example of a modulated signal.

As shown in FIG. 21, the AM transmitter 403 includes an AM subsystem409, a carrier source 413, an information signal source 416, and anamplifier 419. In addition, the AM transmitter 403 may comprise otherelements as well. According to one embodiment, the AM transmitter 403 isone example of amplitude modulation circuitry that is employed togenerate an AM signal as will described. The AM subsystem 409 receivesboth an information signal 423 from the information signal source 416and a carrier signal 426 from the carrier source 413. The carrier source413 may comprise, for example, and oscillator or other source thatgenerates the carrier signal 426 at a desired transmission frequency.

The information signal 423 is used to modulate the carrier signal 426.The information signal source 416 that generates the information signal423 may be, for example, a microphone, audio reproduction equipment, orother source as can be appreciated. The information signal source 416may include various components to provide for signal processing orconditioning such as noise reduction, equalization, pre-amplification,or other signal processing as can be appreciated. The information signal423 may also be called a “modulation signal” since it is used tomodulate the carrier signal 426.

The AM subsystem 409 generates an AM signal 429 from the informationsignal 423 and the carrier signal 426. To this end, the AM subsystem 409may implement any one of a number of different types of amplitudemodulation. Such types of amplitude modulation may comprise, forexample, double-sideband full carrier, single-sideband reduced-carrier,single-sideband full-carrier, single-sideband suppressed-carrier,independent-sideband emission, vestigial-sideband, linked compressor andexpander, or other types of amplitude modulation.

The AM subsystem 409 may be an analog or digital circuit. For example,the AM subsystem 409 may employ digital signal processing to generatethe AM signal 429. To this end, analog-to-digital conversion may beperformed on the information signal 423 and the carrier signal 2019within the modulation subsystem 409 unless such signals are already indigital form before being applied to the modulation subsystem 409.Alternatively, the modulation subsystem 409 may comprise analogcomponents where the carrier signal 426 and the information signal 423are analog signals.

The AM signal 429 may be applied to an amplifier 419 where it isamplified, thereby generating an output AM signal 433. Alternatively,some other components might be used instead of the amplifier 419 togenerate a desired output AM signal 433. The output AM signal 433 isthen sent to the guided surface waveguide probe P through the matchingnetwork 406. To this end, the matching network 406 may comprise one ormore circuit elements configured to facilitate coupling the output AMsignal 433 to the guided surface waveguide probe P while minimizingreflection as can be appreciated. According to one embodiment, the AMtransmitter 403/matching network 406 may be coupled directly to theguided surface waveguide probe P by way of a direct coupling.Alternatively, the AM transmitter 403/matching network 406 may beinductively coupled or link-coupled to the guided surface waveguideprobe P as will be described. It should be noted that the matchingnetwork 406 is employed to minimize or eliminate unwanted reflection inthe system. To this end, when the AM transmitter 403 is coupled to anyembodiment of a guided surface waveguide probe as described herein,various approaches used for impedance matching should be considered toeliminate unwanted reflection.

Next, the general operation of the AM transmission system 400 isdescribed. To begin, the carrier source 413 generates the carrier signal426, and the information signal source 416 generates the informationsignal 423. The information signal 423 and the carrier signal 426 areboth applied to the modulation subsystem 409. The modulation subsystem409 generates the AM signal 429 that is amplified by the amplifier 419,thereby generating the output AM signal 433. The output AM signal 433 isthen applied to the guided surface waveguide probe P through thematching network 406. In some cases, the matching network 406 may not benecessary as will be described. Also, it may be understood that thematching network 406 may be incorporated as part of the AM transmitter403.

The guided surface waveguide probe P is adjusted to launch a guidedsurface wave that embodies the amplitude modulated signal along thesurface of the ground which is also termed a “terrestrial medium” asdescribed above. The guided surface wave propagates along the interfacebetween the terrestrial medium and the atmospheric medium as describedabove. To launch the guided surface wave, the guided surface waveguideprobe P is adjusted to generate a resultant field that provides for thecomplex angle of incidence at least at a Hankel crossover distance fromthe guided surface waveguide probe P as described above. The complexangle of incidence may be calculated as described above.

According to the various embodiments, the guided surface wave thatembodies the amplitude modulated signal decays exponentially as afunction of distance from the guided surface waveguide probe P. In thisrespect, the guided surface wave that embodies the amplitude modulatedsignal is a transmission line mode. Consequently, receivers such as anAM radio will present an electrical load to the AM transmitter 403coupled to the guided surface waveguide probe P. However, given that thetypical AM radio is locally powered, the electrical load that such areceiver will present to the AM transmitter 403 is very small if notnegligible.

In fact, it is believed that the electrical load presented by a given AMradio would be measured in terms of Picowatts. Given this reality, itwould not be possible to come up with enough receivers to present muchof a change in the electrical load experienced by the AM transmitter 403by AM radios. Thus, ultimately, the electrical load seen by AMtransmitter 403 and the guided surface waveguide probe P is likely to berelatively constant and primarily involves resistive losses in theground, guided surface waveguide probe P, the AM transmitter 403, andany other component in the overall system.

Over time, the ground may be subject to change due to changing weatherconditions and the like. For example, dry ground due to sun exposuremight become wet due to a passing storm. As such, the conductivity andpermittivity of the ground will change. Given that the electrical loadexperienced by the guided surface waveguide probe P is not likely tofluctuate significantly, according to various embodiments, the guidedsurface waveguide probe P may include control systems to adjust theoperation of the guided surface waveguide probe P itself for optimaltransmission as will be described.

Alternatively, in some embodiments, the guided surface waveguide probe Pmay be static in nature without control systems to adjust the operationof the guided surface waveguide probe P itself for optimal transmissionrelative to changing conditions on the ground, etc. In such anembodiment, it may be the case that efficiency in transmitting a guidedsurface wave may be affected by changing conductivity and permittivityof the ground. However, in some environments the changes to theconductivity and permittivity of the ground may not be that significantfrom one weather extreme (rain or snow) to another (dry conditions) toaffect the operation of the guided surface waveguide probe Psignificantly. To the extent that any inefficiencies that are introducedby changing weather conditions, such inefficiencies may be toleratedwhere a static guided surface waveguide probe P is employed.

According to one embodiment, the amplifier 419 or other appropriatecomponent of the AM transmitter 403 is controlled to drive the output AMsignal 433 in a manner so as to maintain a constant voltage on an uppercharge terminal T₁ (FIG. 7) of the guided surface waveguide probe P.Since the strength of the resulting electromagnetic field embodied inthe guided surface wave launched by the guided surface waveguide probe Pis proportional to the voltage on the charge terminal T₁, the voltagelevel maintained will depend upon the ultimate desired strength of theelectromagnetic field of the guided surface wave. Note that where theconductivity and permittivity of the ground changes, the efficiency ofthe guided surface waveguide probe P in launching the guided surfacewave may increase or decrease accordingly. As such, the a control systemmay be configured to maintain optimum launching efficiency with respectto launching the guided surface wave so that energy is not lost bydriving the output AM signal 433 unnecessarily with more power via theamplifier 419. In other words, if the launching efficiency of the guidedsurface waveguide probe P is compromised and the amplifier 419 worksharder to maintain the desired voltage and field strength, then it maybe the case that energy is unnecessarily being wasted in the system.

To help maintain optimal operation, a feedback signal may be generatedbased on the magnitude of power produced by the amplifier 419 in drivingthe guided surface waveguide probe P, where the expected electrical loadexperienced by the guided surface waveguide probe P is to remainconstant as is the case with the amplitude modulation transmissionsystem 400 described herein. If the amount of power supplied to theguided surface waveguide probe P exceeds expected values, then thelaunching efficiency may not be optimum and adjustment of the operationof the guided surface waveguide probe P may be desirable. Specificadjustments that may be made with respect to the operation of the guidedsurface waveguide probe P are discussed with reference to later figures.

Due to the fact that the guided surface waveguide probe P launches aguided surface wave that is a transmission line mode, theelectromagnetic field of the guided surface wave decays exponentially asa function of distance from the guided surface waveguide probe P. Inaddition, the guided surface wave does not spread hemispherically, butis bound to the terrestrial medium and spreads as a function ofe^(−ad)/√{square root over (d)} (where d=distance) and exhibits adistinctive knee on the log-log scale. The fact that the guided surfacewave generated by a guided surface waveguide probe P decaysexponentially and is not subject to hemispherical spreading means thatthe field strength before the knee of the curve is much greater than onewould achieve with a comparable conventional quarter-wavelength antenna.The fact that the guided surface wave that embodies the amplitudemodulated signal has such a shape means that the electromagnetic fieldsgenerated by the guided surface waveguide probe P are much stronger thanradiated fields generated by a conventional quarter-wavelength antenna.At some distances, the field strength of the guided surface wave may be100 to 1000 times stronger (or more) than such radiated fields.

In addition, one may recall that in order to generate a radiated fieldusing a conventional antenna, radiation resistance is maximized. Theelectrical length of such an antenna is specified so as to maximizeradiation resistance. One convention is to use an antenna that is atleast a quarter-wavelength in height.

However, according to various embodiments, the electrical length of theguided surface waveguide probe P is electrically small so as to minimizeradiation resistance, which in turn minimizes or substantiallyeliminates the generation of radiated electromagnetic waves by thestructure. According to one embodiment, the electrical length of theguided surface waveguide probe P is less than 1/10^(th) the height of aconventional quarter-wavelength antenna operating at the same frequency,although other electrical lengths may be employed. Assuming that aguided surface waveguide probe P is configured to minimize radiationresistance, then such a structure provides a significant advantage inthat the guided surface waveguide probe P may be much smaller in sizethan a traditional antenna. This translates into a smaller area requiredto mount the guided surface waveguide probe P when considering the needfor guy wires and the like to hold a larger antenna structure in place.

In one further embodiment, it would be possible for the electricalheight of the guided surface waveguide probe P to be specified so thatthe amplitude modulated signal 429 is embodied in both a guided surfacewave launched and a radiated field transmitted by the a modified versionof a guided surface waveguide probe P.

With reference to FIG. 22, shown is another example of a modulatedtransmission system comprising an Amplitude Modulation (AM) transmissionsystem 400 a according to one embodiment of the present disclosure. TheAM transmission system 400 a is a specific example of the AMtransmission system 400 (FIG. 21) described above. The AM transmissionsystem 400 a includes the AM transmitter 403, the matching network 406,and a guided surface waveguide probe P₁ that is similar to the guidedsurface waveguide probe 200 b (FIG. 7). The AM transmitter 403 is thesame as was described with reference to FIG. 21 and includes the carriersource 413, the information signal source 416, the AM subsystem 409, theamplifier 419, and potentially other components as described above.Accordingly, a detailed discussion of the operation of the AMtransmitter 403 is not provided with reference to FIG. 22 as the same isset forth in the discussion of FIG. 21 above.

The AM transmitter 403 is coupled to a matching network 406, which inturn, is directly coupled to the guided surface waveguide probe P₁.Alternatively, the matching network 406 may be inductively coupled orlink-coupled to the guided surface waveguide probe P₁.

According to one embodiment, guided surface waveguide probe P₁ is asingle phase probe having a single charge terminal T₁. The guidedsurface waveguide probe P₁ includes a coil 453. One end of the coil 453is coupled to the charge terminal T₁, and the other end of the coil 453is coupled to a ground stake 456 or other grounding mechanism. The coil453 provides one example of a phase delay circuit that may be used inthe feed network of the guided surface waveguide probe P₁. It isunderstood that other phase delay circuits may be employed as mentionedabove.

Next, the general operation of the AM transmission system 400 a isdescribed. To begin, the AM transmitter 403 generates an output AMsignal 433 that is applied to the guided surface waveguide probe P₁through the matching network 406. As mentioned above, in some situationsthe matching network 406 may not be necessary where the output impedanceof the AM transmitter 403 is impedance matched to the guided surfacewaveguide probe P₁. Specifically, if the output impedance of the AMtransmitter 403 is known, the output of the AM transmitter 403 may becoupled to tap at an appropriate location on the coil 453 to match theoutput impedance of the AM transmitter 403 to eliminate unwantedreflections.

The guided surface waveguide probe P₁ is adjusted to launch a guidedsurface wave that embodies the AM signal along the surface of the groundor terrestrial medium as described above. To this end, the guidedsurface wave propagates along the interface between the terrestrialmedium and the atmospheric medium as described above. The guided surfacewaveguide probe P₁ is adjusted to generate a resultant field thatprovides for the complex angle of incidence at least at a Hankelcrossover distance from the guided surface waveguide probe P₁ asdescribed above. Stated further, the charge terminal T₁ elevated overthe terrestrial medium or the lossy conducting medium is configured togenerate at least one resultant field that synthesizes a wave frontincident at the complex Brewster angle of incidence (θ_(i,B)) of theterrestrial medium or the lossy conducting medium. The complex angle ofincidence or complex Brewster angle of incidence may be calculated asdescribed above.

The coil 453 and feed lines between the coil 453 (and the switchingmechanism 364) and the charge terminal T₁ comprise a feed network thatis electrically coupled to the charge terminal T₁. This feed networkprovides for a phase delay (Φ) that matches a wave tilt angle (Ψ)associated with a complex Brewster angle of incidence (θ_(i,B))associated with the terrestrial medium (which is a lossy conductingmedium) in the vicinity of the guided surface waveguide probe P₁.

As was discussed above, the guided surface wave that embodies theamplitude modulated signal decays exponentially as a function ofdistance from the guided surface waveguide probe P₁. In this respect,the guided surface wave that embodies the amplitude modulated signal isa transmission line mode. Consequently, receivers such as an AM radiowill present an electrical load to the AM transmitter 403 and the guidedsurface waveguide probe P₁. However, given that the typical AM radio islocally powered, the electrical load that such a receiver will presentto the guided surface waveguide probe P₁ is very small if not negligibleas mentioned above. Ultimately, the electrical load seen by the guidedsurface waveguide probe P₁ is likely to be very small and relativelyconstant as was described above with respect to the guided surfacewaveguide probe P (FIG. 21).

As was mentioned above, over time the ground or terrestrial medium maybe subject to change due to changing weather conditions and the like.For example, dry ground due to sun exposure might become wet due to apassing storm. As such, the conductivity and permittivity of the groundwill change. It is possible that the efficiency of the operation of theguided surface waveguide probe P₁ may be effected by the changingconductivity and permittivity in such cases. However, in somesituations, the effect of changing conductivity and permittivity on theoperation of the guided surface waveguide probe P₁ may be within anacceptable or tolerable limits. As such, the AM transmitter 403 maycompensate for any efficiency by providing more power to the system. Tothis end a feedback signal may be generated from an ammeter at the baseof the guided surface waveguide probe P₁ of from a field meter near thesite of transmission that is provided to the AM transmitter 403 so thatthe AM transmitter 403 can work to maintain a constant voltage on theterminal T₁ despite changing conditions around the site of transmission.In any event, according to one embodiment, amplifier 419 or otherappropriate component of the AM transmitter 403 is controlled to drivethe output AM signal 433 in a manner so as to maintain a constantvoltage on the charge terminal T₁.

With reference to FIG. 23, shown is yet another example of a modulatedtransmission system comprising an Amplitude Modulation (AM) transmissionsystem 400 b according to one embodiment of the present disclosure. TheAM transmission system 400 b is another specific example of the AMtransmission system 400 (FIG. 21) described above. The AM transmissionsystem 400 b includes the AM transmitter 403, the matching network 406,and a guided surface waveguide probe P₂ that is similar to the guidedsurface waveguide probe 200 b (FIG. 7) with some differences as will bedescribed below. The AM transmitter 403 is the same as was describedwith reference to FIG. 21 and includes the carrier source 413, theinformation signal source 416, the AM subsystem 409, the amplifier 419,and potentially other components as described above. Accordingly, adetailed discussion of the operation of the AM transmitter 403 is notprovided with reference to FIG. 23 as the same is set forth in thediscussion of FIG. 21 above.

The AM transmitter 403 is coupled to a matching network 406, which inturn, is directly coupled to the guided surface waveguide probe P₂.Alternatively, the matching network 406 may be inductively coupled orlink-coupled to the guided surface waveguide probe P₂.

According to one embodiment, guided surface waveguide probe P₂ is asingle phase probe having a single charge terminal T₁ similar to theguided surface waveguide probe P₁ (FIG. 22). To this end, the guidedsurface waveguide probe P₂ includes the coil 453. One end of the coil453 is coupled to the charge terminal T₁, and the other end of the coil453 is coupled to the ground stake 456 or other grounding mechanism. Anammeter 459 is positioned at the ground stake 456 to measure the currentat that point. The coil 453 provides one example of a phase delaycircuit that may be used in the feed network of the guided surfacewaveguide probe P₂. It is understood that other phase delay circuits maybe employed as mentioned above.

The guided surface waveguide probe P₂ further includes a switchingmechanism 463. The switching mechanism 463 may comprise, for example,roller in a roller inductor, a solid state device such as a multiplexer,a mechanical switching device, or other configuration. Where theswitching mechanism is a roller in a roller inductor, the coil 453 ispart of the roller inductor as will be described in further detailbelow. The specific type of switching mechanism 463 used will dependupon the voltage and current handling requirements for a given guidedsurface waveguide probe P (FIG. 21). An input of the switching mechanism463 is coupled to the charge terminal T₁. The switching mechanism 463has a plurality of outputs coupled to taps on the coil 453. At any giventime, depending on its state, the switching mechanism 463 couples thecharge terminal T₁ to a respective one of the taps, thereby coupling thecharge terminal T₁ to a respective point on the coil 453. In thismanner, the phase delay due to the coil 453 may be adjusted to providefor optimal operation of the guided surface waveguide probe P₂ dependingon the physical parameters of the ground that may change from time totime due to weather or other factors.

To this end, the guided surface waveguide probe P₂ also includes awaveguide probe control system 466. The waveguide probe control system466 controls the position of the switching mechanism 463 to optimize theoperation of the guided surface waveguide probe P₂ as will be described.A control output from the waveguide probe control system 466 is coupledto a control input of the switching mechanism 463 and allows thewaveguide probe control system 466 to direct the state of the switchingmechanism 463 to determine the tap to which the charge terminal T₁ isconnected at any given time.

It should be noted that a field meter 469 may also be coupled to thewaveguide probe control system 466 to provide field strength readingsthereto. According to various embodiments, the field meter 469 may beemployed in place of the ammeter 459 or in addition to the ammeter 459.

Next, the general operation of the AM transmission system 400 b isdescribed. To begin, the AM transmitter 403 generates an output AMsignal 433 that is applied to the guided surface waveguide probe P₂through the matching network 406. As mentioned above, the matchingnetwork 406 may not be necessary.

The guided surface waveguide probe P₂ is adjusted to launch a guidedsurface wave that embodies the AM signal along the surface of the groundor terrestrial medium as described above. To this end, the guidedsurface wave propagates along the interface between the terrestrialmedium and the atmospheric medium as described above. The guided surfacewaveguide probe P₂ is adjusted to generate a resultant field thatprovides for the complex angle of incidence at least at a Hankelcrossover distance from the guided surface waveguide probe P₂ asdescribed above. Stated further, the charge terminal T₁ elevated overthe terrestrial medium or the lossy conducting medium is configured togenerate at least one resultant field that synthesizes a wave frontincident at the complex Brewster angle of incidence (θ_(i,B)) of theterrestrial medium or the lossy conducting medium. The complex angle ofincidence or complex Brewster angle of incidence may be calculated asdescribed above.

Alternatively, the coil 453 and feed lines between the coil 453 (and theswitching mechanism 364) and the charge terminal T₁ comprise a feednetwork that is electrically coupled to the charge terminal T₁. Thisfeed network provides for a phase delay (Φ) that matches a wave tiltangle (Ψ) associated with a complex Brewster angle of incidence(θ_(i,B)) associated with the terrestrial medium (which is a lossyconducting medium) in the vicinity of the guided surface waveguide probeP₂.

As was discussed above, the guided surface wave that embodies theamplitude modulated signal decays exponentially as a function ofdistance from the guided surface waveguide probe P₂. In this respect,the guided surface wave that embodies the amplitude modulated signal isa transmission line mode. Consequently, receivers such as an AM radiowill present an electrical load to the AM transmitter 403 and the guidedsurface waveguide probe P₂. However, given that the typical AM radio islocally powered, the electrical load that such a receiver will presentto the guided surface waveguide probe P₂ is very small if not negligibleas mentioned above. Ultimately, the electrical load seen by the guidedsurface waveguide probe P₂ is likely to be very small and relativelyconstant as was described above with respect to the guided surfacewaveguide probe P (FIG. 21).

As was mentioned above, over time the ground or terrestrial medium maybe subject to change due to changing weather conditions and the like.For example, dry ground due to sun exposure might become wet due to apassing storm. As such, the conductivity and permittivity of the groundwill change. Given that the electrical load experienced by the guidedsurface waveguide probe P₂ is not likely to fluctuate significantly, thewaveguide probe control system 466 can be configured to adjust the tapsby controlling the switching mechanism 463 using the current at theground stake 456 as detected by the ammeter 459, or the field strengthas detected by the field meter 469, as feedback in order to optimize thelaunching of the guided surface wave.

Ultimately, the taps may be adjusted to optimize the operation of theguided surface waveguide probe P₂. According to one embodiment, theoperation of the guided surface waveguide probe P₂ is optimized when thecurrent at the ground stake 456 is maximized or when the field strengthof the guided surface wave generated by the guided surface waveguideprobe is maximized. Thus, in one embodiment, the taps may be adjusted inorder to maximize current at the ground stake 456 as detected by theammeter 459 or to maximize the field strength measured by the fieldmeter 469 to ensure optimal operation in view of changing groundconditions or other conditions. As an alternative, the waveguide probecontrol system 466 may obtain a feedback signal from a ground parametermeter that provides current values of conductivity and permittivity forthe ground local to the guided surface waveguide probe P₂ as describedabove. The guided surface waveguide probe P₂ may be adjusted tosubstantially match an optimal configuration of the guided surfacewaveguide probe P₂ as calculated from the values of conductivity andpermittivity of the ground at the site of the guided surface waveguideprobe P₂.

According to one embodiment, amplifier 419 or other appropriatecomponent of the AM transmitter 403 is controlled to drive the output AMsignal 433 in a manner so as to maintain a constant voltage on thecharge terminal T₁. Since the strength of the resulting electromagneticfield embodied in the guided surface wave launched by the guided surfacewaveguide probe P₂ is proportional to the voltage on the charge terminalT₁, the voltage level maintained will depend upon the ultimate desiredstrength of the electromagnetic field of the guided surface wave. Notethat where the conductivity and permittivity of the ground changes, theefficiency of the guided surface waveguide probe P₂ in launching theguided surface wave may increase or decrease accordingly. As such, thewaveguide probe control system 466 is configured to maintain optimumlaunching efficiency with respect to launching the guided surface waveso that energy is not lost by driving the output AM signal 433 with moreenergy that is necessary via the amplifier 419. In other words, if thelaunching efficiency of the guided surface waveguide probe P₂ iscompromised and the AM transmitter 403 is forced to harder (via theamplifier) to maintain the desired voltage and field strength, then itmay be the case that energy is unnecessarily being wasted in the system.To this end, it is also possible that a feedback signal may be generatedbased on the magnitude of power output produced by the AM transmitter403 in driving the guided surface waveguide probe P₂, where the expectedelectrical load experienced by the guided surface waveguide probe P₂ isto remain constant as is the case with the amplitude modulationtransmission system 400 described herein. If the amount of powersupplied to the guided surface waveguide probe P₂ exceeds expectedvalues, then the launching efficiency may not be optimum and adjustmentof the taps via the switching mechanism 463 may be desirable.Alternatively, other parameters of the guided surface waveguide probe P₂might be adjusted. Note that a signal may be sent from the AMtransmitter 403 to the waveguide probe control system 466 to provide anindication as to whether energy is being wasted due to the fact that theAM transmitter 403 is overdriving the guided surface waveguide probe P₂.

With reference to FIG. 24, shown is yet another example of a modulatedtransmission system comprising an Amplitude Modulation (AM) transmissionsystem 400 c according to one embodiment of the present disclosure. TheAM transmission system 400 c is another example of the AM transmissionsystem 400 (FIG. 21) described above. The AM transmission system 400 cincludes the AM transmitter 403, the matching network 406, and a guidedsurface waveguide probe P₃ that is similar to the guided surfacewaveguide probe 200 d (FIG. 14). The AM transmitter 403 is the same aswas described with reference to FIG. 21 and includes the carrier source413, the information signal source 416, the AM subsystem 409, theamplifier 419, and potentially other components as described above.Accordingly, a detailed discussion of the operation of the AMtransmitter 403 is not provided with reference to FIG. 24 as the same isset forth in the discussion of FIG. 21 above.

According to one embodiment, guided surface waveguide probe P₃ is asingle phase probe having a single charge terminal T₁ with acompensation terminal 473. The guided surface waveguide probe P₃includes the coil 453. One end of the coil 453 is coupled to the chargeterminal T₁, and the other end of the coil 453 is coupled to the groundstake 456 or other grounding mechanism. An ammeter 459 is positioned atthe ground stake 456 to measure the current at that point. The coil 453provides one example of a phase delay circuit that may be used in thefeed network of the guided surface waveguide probe P₃. It is understoodthat other phase delay circuits may be employed as mentioned above.

The guided surface waveguide probe P₃ further includes a switchingmechanism 476 that is similar to the switching mechanism 463 discussedabove with reference to FIG. 23. However, the guided surface waveguideprobe P₃ differs from the guided surface waveguide probe P₂ (FIG. 23) inthat an input of the switching mechanism 463 is coupled to thecompensation terminal 473. The switching mechanism 2049 has a pluralityof outputs coupled to taps on the coil 453. At any given time, dependingon its state, the switching mechanism 463 couples the compensationterminal 473 to a respective one of the taps.

The guided surface waveguide probe P₃ also includes a waveguide probecontrol system 479. The waveguide probe control system 479 controls theposition of the switching mechanism 463 to optimize the operation of theguided surface waveguide probe P₃ in a similar manner to the waveguideprobe control system 466 (FIG. 23) described above. To this end, acontrol output from the waveguide probe control system 479 is coupled toa control input of the switching mechanism 476 and allows the waveguideprobe control system 479 to direct the state of the switching mechanism476 to determine the tap to which the compensation terminal 473 isconnected at any given time.

The guided surface waveguide probe P₃ is adjusted to launch a guidedsurface wave that embodies the amplitude modulated signal along thesurface of the ground or terrestrial medium as described above. To thisend, the guided surface wave propagates along the interface between theterrestrial medium and the atmospheric medium as described above. Theguided surface waveguide probe P₃ is adjusted to generate a resultantfield that provides for the complex angle of incidence at least at aHankel crossover distance from the guided surface waveguide probe P₃ asdescribed above. Stated further, the charge terminal T₁ elevated overthe terrestrial medium or the lossy conducting medium is configured togenerate at least one resultant field that synthesizes a wave frontincident at the complex Brewster angle of incidence (θ_(i,B)) of theterrestrial medium or the lossy conducting medium. The complex angle ofincidence or complex Brewster angle of incidence may be calculated asdescribed above.

According to the various embodiments, the guided surface wave thatembodies the amplitude modulated signal decays exponentially as afunction of distance from the guided surface waveguide probe P₃. In thisrespect, the guided surface wave that embodies the amplitude modulatedsignal is a transmission line mode. Consequently, receivers such as anAM radio will load the guided surface waveguide probe P₃ and the AMtransmitter 403 coupled thereto through the matching network 406.However, given that the typical AM radio is locally powered, theelectrical load that such a receiver will present to the guided surfacewaveguide probe P₃/AM Transmitter 403 is very small if not negligible aswas discussed above. Given this reality, it would not be possible tocome up with enough receivers to present much of a change in theelectrical load presented by AM receivers for the AM transmitter 402 andthe guided surface waveguide probe P₃. Thus, ultimately, the electricalload seen by the AM transmitter 403 and the guided surface waveguideprobe P₃ is likely to be relatively constant and primarily involvesresistive losses in the ground and in the components that make up the AMtransmitter 403, the matching network 406, and the guided surfacewaveguide probe P₃ itself.

Given that the electrical load experienced by the guided surfacewaveguide probe 2006 a is not likely to fluctuate significantly, as theconductivity and permittivity of the ground changes over time asdiscussed above, the waveguide probe control system 479 can beconfigured to adjust the taps by controlling the switching mechanism 476using the current at the ground stake 456 as detected by the ammeter 459as feedback in order to optimize the launching of the guided surfacewave. The taps may be adjusted in order to maximize current at theground stake 456 to ensure optimal operation in view of changing groundconditions. In embodiments where field readings are provided by thefield meter 469 as feedback to the waveguide probe control system 479,the taps may be adjusted to maintain a predefined field strength of thefields created by the guided surface wave.

As an alternative, the waveguide probe control system 479 may obtain afeedback signal from a ground parameter meter that provides real timevalues of conductivity and permittivity for the ground local to theguided surface waveguide probe P₃ as described above. The waveguideprobe control system 479 may select a tap so that the physicalconfiguration of the guided surface waveguide probe P₃ substantiallymatches an optimal configuration of the guided surface waveguide probeP₃ as calculated from the values of conductivity and permittivity of theground at the site of the guided surface waveguide probe P₃.

According to one embodiment, amplifier 419 or other appropriatecomponent of the AM transmitter 403 is controlled to drive the output AMsignal 433 in a manner so as to maintain a constant voltage on thecharge terminal T₁. Since the strength of the resulting electromagneticfield embodied in the guided surface wave launched by the guided surfacewaveguide probe P₃ is proportional to the voltage on the charge terminalthe voltage level maintained will depend upon the ultimate desiredstrength of the electromagnetic field of the guided surface wave. Notethat where the conductivity and permittivity of the ground changes, theefficiency of the guided surface waveguide probe P₃ in launching theguided surface wave may increase or decrease accordingly. As such, thewaveguide probe control system 479 is configured to maintain optimumlaunching efficiency with respect to launching the guided surface waveso that energy is not lost by driving the output AM signal 433 with morepower via the amplifier 419 than should be necessary to maintain thedesired field strengths. In other words, if the launching efficiency ofthe guided surface waveguide probe P₃ is compromised and the amplifier419 works harder to maintain the desired voltage and field strength,then it may be the case that energy is unnecessarily being wasted in thesystem. To this end, it is also possible that a feedback signal may begenerated based on the magnitude of power, current, or other parameterindicating a degree of the output produced by the amplifier 419 or othercomponent in driving the guided surface waveguide probe P₃, where theexpected electrical load experienced by the guided surface waveguideprobe P₃ and the AM transmitter 403 is to remain relatively constant. Ifthe amount of power supplied to the guided surface waveguide probe P₃and the AM transmitter 403 exceeds expected values, then the launchingefficiency may not be optimum and adjustment of the taps via theswitching mechanism 476 may be desirable. Alternatively, otherparameters of the guided surface waveguide probe P₃ might be adjusted.

Referring next to FIG. 25, shown is still another example of a modulatedtransmission system comprising an Amplitude Modulation (AM) transmissionsystem 400 d according to one embodiment of the present disclosure. TheAM transmission system 400 d is yet another example of the AMtransmission system 400 (FIG. 21) described above. The AM transmissionsystem 400 d includes the AM transmitter 403, the matching network 406,and a guided surface waveguide probe P₄ that is similar to the guidedsurface waveguide probe 200 f (FIG. 17). The AM transmitter 403 is thesame as was described with reference to FIG. 21 and includes the carriersource 413, the information signal source 416, the AM subsystem 409, theamplifier 419, and potentially other components as described above.Accordingly, a detailed discussion of the operation of the AMtransmitter 403 is not provided with reference to FIG. 25 as the same isset forth in the discussion of FIG. 21 above.

The guided surface waveguide probe P₄ is a poly-phase probe thatincludes a first charge terminal T₁ and a second charge terminal T₂. Theguided surface waveguide probe P₄ further includes a roller mechanism483. The guided surface waveguide probe P₄ further includes a coil 486.The roller mechanism 483 contacts the coil 486 in a continuous mannerand varies its position as can be appreciated so that the position wherethe charge terminal T₁ is electrically coupled to the coil 486 may varyaccordingly.

The roller mechanism 483 is part of a roller inductor that is oneexample of a variable inductor that may be used. Alternatively, aswitching mechanism 463 (FIG. 23) as described above may be employed. Afree end of the coil 486 is coupled to the second charge terminal T₂.The input of the roller mechanism 483 is coupled to the first chargeterminal T₁. As an alternative, the input of the roller mechanism 483may be coupled to the second charge terminal T₂ and a free end of thecoil 486 may be coupled to the first charge terminal T₁. In the end, theuse of the roller mechanism 483 facilitates varying the length of thecoil 486 coupled between the first charge terminal T₁ and the secondcharge terminal T₂. The coil 486 provides one example of a phase delaycircuit that may be used in the feed network of the guided surfacewaveguide probe P₄. It is understood that other phase delay circuits maybe employed as mentioned above.

The guided surface waveguide probe P₄ includes a waveguide probe controlsystem 489. The waveguide probe control system 489 generates a controloutput that is applied to control the position of the roller mechanism483 on the coil 486. In addition, the signal output of a field meter 469may be coupled to a feedback input of the waveguide probe control system489. Alternatively, the signal output of a ground parameter meter 493may be coupled to a feedback input of the waveguide probe control system489.

According to one embodiment, the AM transmitter 403 is inductivelycoupled to the guided surface waveguide probe P₄ through matchingnetwork 406. However, it is understood that the AM transmitter 403 andmatching network 406 may be directly coupled to the guided surfacewaveguide probe P₄, where such inductive coupling to the guided surfacewaveguide probe P₄ is shown as an example.

The amplitude modulation transmission system 400 d operates in a mannersimilar to the amplitude modulation transmission system 400 (FIG. 21)described above. To this end, the amplifier 419 or other appropriatecomponent of the AM transmitter 403 drives the output AM signal 433 thatis applied to the guided surface waveguide probe P₄ through the matchingnetwork 406. In one embodiment, the waveguide probe control system 489receives feedback from the field meter 469 and operates to generate aguided surface wave with desired field strengths. Alternatively, thewaveguide probe control system 489 may obtain a signals from a groundparameter meter 493 that provides current values of conductivity andpermittivity for the ground local to the guided surface waveguide probeP₄ as described above. The guided surface waveguide probe P₄ may beadjusted to substantially match an optimal configuration of the guidedsurface waveguide probe P₄ as calculated from the values of conductivityand permittivity of the ground at the site of the guided surfacewaveguide probe P₄ as set forth above. In addition, the waveguide probecontrol system 489 may receive a feedback signal generated from thepower output of the amplifier 419 or other appropriate component of theAM transmitter 403 that indicates the power supplied to the guidedsurface waveguide probe P₄.

Based on one or more of these feedback parameters, the waveguide probecontrol system 489 controls the position of the roller mechanism 483 tomaximize the launching efficiency of the guided surface wave and,correspondingly, minimize the magnitude of the power supplied by theamplifier 419 while maintaining a desired voltage on the first or uppercharge terminal T₁ and/or the desired field strength associated with theresulting guided surface wave launched. The waveguide probe controlsystem 489 may further take into account that the electrical loadperceived by the guided surface waveguide probe P₄ is likely to benegligible and relatively constant as described above.

With reference to FIG. 26, shown is an illustration of skywavepropagation from a conventional antenna. Specifically, as radiatedelectromagnetic fields propagate from a source antenna 503, they willrefract off of the ionization layer 506 in the sky and propagate back tothe ground some distance from the source antenna 503. In this respect,there is a skywave propagation path 509 and a ground wave propagationpath 513 associated with a skywave. Generally, this effect is morepronounced at night given the effect of solar radiation on theionization layer during the day. Also shown is the virtual height H_(V)of the ionization layer 506.

According to the various embodiments, the guided surface waveguideprobes described herein are anti-skywave structures. That is to say, theguided surface waveguide probes P launch the guided surface wavesembodying modulated signals as described herein while minimizing oreliminating the creation of any skywave. This is due to the nature ofthe guided surface waves in that they are bound to the surface of theearth or terrestrial medium. Thus, modulated signals such as AM signalsmay be transmitted using a guided surface waveguide probe without theconcern of creating problematic skywaves. To the extent that anyradiated fields are created by a given guided surface waveguide probe P,such radiated fields are likely to be relatively small or negligibleprovided that the guided surface waveguide probe P is electrically smallas described above. As a consequence, one may transmit a modulatedsignal in the form of a guided surface wave as described herein withouthaving to reduce the power of transmission at nighttime to preventskywave interference with other stations.

In addition, with reference to the various embodiments described abovein FIGS. 21-26, various shielding techniques may be employed to furtherreduce any radiation, however minimal, that might be generated by thevarious embodiments of the guided surface waveguide probes P describedherein.

Referring next to FIG. 27, shown is a drawing that provides anillustration of first and second guided surface wave (GSW) fieldstrength curves 2096 a and 2096 b associated with of a pair of guidedsurface waves generated by a corresponding pair of guided surfacewaveguide probes as described above. The GSW field strength curves 2096a/b show field strengths as a function of distance as can beappreciated. Also depicted are radiated field strength curves 2097 a/bthat show example field strengths of radiated fields generated byconventional antennas. To this end, the respective guided surfacewaveguide probes are positioned so that their areas of transmissionoverlap in a predefined manner. According to various embodiments, thefield strengths and physical separation of the respective guided surfacewaveguide probes are specified so that a predefined overlap is achievedsuch that a first AM signal embodied in the first guided surface wavedoes not substantially interfere with a second AM signal embodied in thesecond guided surface wave.

As mentioned above, each of the GSW field strength curves 2096 a/b ofthe respective guided surface waves decay exponentially as a function ofe^(−ad)/√{square root over (d)} (see above) where d is the distance fromthe respective guided surface waveguide probes. This is because suchguided surface waves comprise transmission line modes as mentionedabove. As such, the GSW field strength curves 2096 a/b each have thecharacteristic knee as described above.

It should be noted that the spreading factor for a radiated field is afunction of 1/r, where r is distance. However, the spreading factor fora guided surface wave is a function of 1/√{square root over (r)}. Aradiated field spreads hemispherically whereas a guided surface wave isbound to the waveguide which is the lossy conducting medium or theground. As a consequence, a guided surface wave has a much smallerspreading factor that translates into a stronger signal before the kneeof the curve defined by the exponential decay. In addition, theexponential decay of the guided surface wave means that there is a moredefined signal cutoff beyond the characteristic knee of the fieldstrength curve as described above. As a consequence, the signal strengthof a guided surface wave is stronger before the knee and the relativelyabrupt cutoff due to the exponential decay means that the possibility ofinterference with other transmitted signals is reduced.

Specifically, due to the fact that the GSW field strength curves 2096a/b associated with guided surface waves launched by a guided surfacewaveguide probe decay exponentially, it is possible to place twodifferent service areas relatively close without substantialinterference. Such is the case even if the frequencies of transmissionare the same for both guided surface waveguide probes.

To illustrate, assume that two different guided surface waveguide probesare configured to transmit guided surface waves embodying AM signals,where the guided surface waves have field strengths that are equal inmagnitude at respective distances from the corresponding guided surfacewaveguide probes. Each GSW field strength curve 2096 a/b includes aservice contour SC that is located at a distance from the originatingguided surface waveguide probe having a field strength at a predefinedthreshold considered appropriate to avoid interference with variousother radio signals, fields generated by local electrical equipment, andother sources of interference. In one embodiment, such a threshold maybe set at 0.5 millivolts per meter (mV/m) other appropriate threshold.This threshold defines a service contour SC of a respective transmittingstructure.

To the extent that adjacent GSW field strength curves 2096 a/b overlap,the second field strength of a second guided surface waveguide probemust be specified so as to have a field strength that is below apredefined threshold at the service contour SC associated with the firstguided surface waveguide probe so as to avoid any substantialinterference with the AM signal represented by the first guided surfacewaveguide probe.

According to one convention, such a predefined threshold is set at1/20th the field strength of the first guided surface waveguide probe.That is to say, in one embodiment, where the field strength of the firstGSW field strength curve 2096 a is equal to 0.5 mV/m, the field strengthof the second GSW field strength curve 2096 b associated with a secondadjacent guided surface waveguide probe must be less than or equal to avalue that is twenty times less than 0.5 mV/m which translates to 0.025mV/m in order to avoid unacceptable interference between the respectivesignals. The point at which a given GSW field strength curve 2096 b hasa strength at the 0.025 mV/m may be termed the interference contour ICof a given guided surface waveguide probe. One advantage, among others,to the use of guided surface waves for the transmission of AM signals isthat the signal strength of such signals for much of a given servicearea is much stronger than can be created via conventional antennas. Assuch, it is possible to increase the signal-to-noise ratio of AMsignals. Given the fact that guided surface waves have thecharacteristic knee, the distance between the service contour SC and theinterference contour IC for a given guided surface waveguide probe ismuch less than the same distance for a radiated field generated by aconventional quarter wave antenna.

As shown in FIG. 27, the fact that the distance between the servicecontour SC and the interference contour IC for a given guided surfacewaveguide probe is much less than the same distance for a radiated fieldgenerated by a conventional quarter wave antenna means that more areamay be covered by respective guided surface waveguide probes and therespective service areas can be closer together without unacceptableinterference.

As shown in FIG. 27, a first guided surface waveguide probe 3003 a ispositioned adjacent to a second guided surface waveguide probe 3003 b.The fact that the distance between the respective service contoursSC_(Sa/b) and their counterpart interference contours IC_(Sa/b) (where“S” refers to “surface” for a guided surface wave) is much reduced, thearea of interference between such adjacent probes is relatively small.This compares favorably with the distance between the service contoursSC_(Ra/b) and interference contours IC_(Ra/b) of radiated fieldsrepresented by the radiated field strength curves 2097 a/b.Specifically, the service contours SCRa/b are much closer to theposition of a corresponding radiating antenna (not shown) located at thesame position as the guided surface waveguide probes 3003 a/b. Thus, theguided surface waveguide probes 3003 a/b provide a much larger area ofservice with much reduced areas of interference relative to conventionalantennas.

Referring next to FIG. 28, shown are service areas 3006 associated withthree different guided surface waveguide probes transmitting guidedsurface waves embodying AM signals. As shown, a network of adjacentservice areas may be specified with distances between service contoursSC and interference contours IC being relatively small with respect tothe entire effective area of service. As such, it would be possible tocreate a honeycomb or other arrangement of service areas associated withguided surface waveguide probes transmitting guided surface wavesembodying AM signals to provide superior coverage of a given area.

It should be emphasized that the above-described embodiments of thepresent disclosure are merely possible examples of implementations setforth for a clear understanding of the principles of the disclosure.Many variations and modifications may be made to the above-describedembodiment(s) without departing substantially from the spirit andprinciples of the disclosure. All such modifications and variations areintended to be included herein within the scope of this disclosure andprotected by the following claims. In addition, all optional andpreferred features and modifications of the described embodiments anddependent claims are usable in all aspects of the disclosure taughtherein. Furthermore, the individual features of the dependent claims, aswell as all optional and preferred features and modifications of thedescribed embodiments, are combinable and interchangeable with oneanother.

Therefore, the following is claimed:
 1. A system, comprising: modulationcircuitry configured to generate a modulated signal; and a guidedsurface waveguide probe configured to launch a Zenneck surface waveembodying the modulated signal along a lossy conducting medium.
 2. Thesystem of claim 1, wherein an electromagnetic field of the Zennecksurface wave decays exponentially as a function of distance from theguided surface waveguide probe.
 3. The system of claim 1, wherein themodulated signal is generated using a carrier signal received from acarrier source and an information signal received from a informationsource.
 4. The system of claim 1, further comprising a control system,the control system being configured to adjust an operation of the guidedsurface waveguide probe based at least in part on one or moreenvironmental conditions.
 5. The system of claim 1, wherein themodulation circuitry further comprises amplitude modulation circuitry,and a modulation type of the modulation performed by the amplitudemodulation circuitry comprises a double-sideband full carrier, asingle-sideband reduced-carrier, a single-sideband full-carrier,single-sideband suppressed-carrier, an independent-sideband emission, avestigial-sideband, or a linked compressor and expander.
 6. The systemof claim 1, further comprising: a feed network electrically coupled to acharge terminal of the guided surface waveguide probe, the feed networkproviding a phase delay (Φ) that matches a wave tilt angle (Ψ)associated with a complex Brewster angle of incidence (θ_(i,B)) of thelossy conducting medium in a vicinity of the guided surface waveguideprobe.
 7. The system of claim 6, wherein the modulation circuitry iscoupled to a phase delay circuit of the feed network.
 8. A method,comprising: generating a modulated signal; and adjusting a guidedsurface waveguide probe to launch a Zenneck surface wave embodying themodulated signal along a lossy conducting medium via a guided surfacewaveguide probe.
 9. The method of claim 8, further comprising couplingthe modulated signal to the guided surface waveguide probe.
 10. Themethod of claim 8, wherein the guided surface waveguide probe comprisesa charge terminal elevated over the lossy conducting medium configuredto generate at least one resultant field that synthesizes a wave frontincident at a complex Brewster angle of incidence (θ_(i,B)) of the lossyconducting medium at a vicinity of the guided surface waveguide probe.11. The method of claim 8, wherein the guided surface waveguide probefurther comprises a feed network electrically coupled to a chargeterminal, the feed network providing a phase delay (Φ) that matches awave tilt angle (Ψ) associated with a complex Brewster angle ofincidence (θ_(i,B)) associated with the lossy conducting medium in avicinity of the guided surface waveguide probe.
 12. The method of claim8, wherein an electromagnetic field of the Zenneck surface wave decaysexponentially as a function of a distance from the guided surfacewaveguide probe.
 13. The method of claim 8, further comprising receivingan information signal from an information source; receiving a carriersignal from a carrier source; and wherein generating the modulatedsignal comprises modulating the carrier signal based at least in part onthe information signal.
 14. An apparatus, comprising: modulationcircuitry configured to generate a modulated signal; and a guidedsurface waveguide probe coupled to the modulation circuitry, the guidedsurface waveguide probe being adjusted to launch a Zenneck surface wavealong a lossy conducting medium, the Zenneck surface wave embodying themodulated signal.
 15. The apparatus of claim 14, further comprising amatching network, the modulation circuitry being coupled to the guidedsurface waveguide probe via the matching network.
 16. The apparatus ofclaim 15, wherein the matching network is employed to minimize creationof a skywave.
 17. The apparatus of claim 14, wherein an electromagneticfield of the guide surface wave decays exponentially as a function ofdistance from the guided surface waveguide probe.
 18. The apparatus ofclaim 14, wherein the modulation circuitry comprises a modulationsubsystem, a carrier source, an information signal source and anamplifier, the modulation subsystem being configured to modulate acarrier signal received from the carrier source using an informationsignal received from the information signal source.
 19. The apparatus ofclaim 14, wherein the guided surface waveguide probe comprises a chargeterminal and a feed network electrically coupled to the charge terminal,and further comprising generating a resultant field that matches a wavetilt angle (Ψ) associated with a complex Brewster angle of incidence(θ_(i,B)) associated with the lossy conducting medium associated withthe guided surface waveguide probe.
 20. The apparatus of claim 19,wherein the modulated signal is an amplitude modulated signal.